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Mathematics · Textbook solutions

Differential Equations

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 176 questions

  1. Find order and degree of the following differential equations.
    i)
    x2d2ydx2+3xdydx+4y=0x^2 \dfrac{d^2y}{dx^2} + 3x \dfrac{dy}{dx} + 4y = 0
  2. ii)
    (d3ydx3)2+xydydx−2x+3y+7=0\left(\dfrac{d^3y}{dx^3}\right)^2 + xy \dfrac{dy}{dx} - 2x + 3y + 7 = 0
  3. iii)
    rdrdθ+cos⁡θ=5r \dfrac{dr}{d\theta} + \cos\theta = 5
  4. iv)
    (d2ydx2)2+(dydx)2=ex\left(\dfrac{d^2y}{dx^2}\right)^2 + \left(\dfrac{dy}{dx}\right)^2 = e^x
  5. v)
    dydx+3xydydx=cos⁡x\dfrac{dy}{dx} + \dfrac{3xy}{\dfrac{dy}{dx}} = \cos x
  6. vi)
    1+1(dydx)2=(d2ydx2)32\sqrt{1 + \dfrac{1}{\left(\dfrac{dy}{dx}\right)^2}} = \left(\dfrac{d^2y}{dx^2}\right)^{\frac{3}{2}}
  7. vii)
    d4ydx4=[1+(dydx)2]3\dfrac{d^4y}{dx^4} = \left[1 + \left(\dfrac{dy}{dx}\right)^2\right]^3
  8. viii)
    edydx+dydx=xe^{\frac{dy}{dx}} + \dfrac{dy}{dx} = x
  9. ix)
    ∣x3y332x23ydydx05x2[yd2ydx2+(dydx)2]0∣=0\begin{vmatrix} x^3 & y^3 & 3 \\ 2x^2 & 3y\dfrac{dy}{dx} & 0 \\ 5x & 2\left[y\dfrac{d^2y}{dx^2} + \left(\dfrac{dy}{dx}\right)^2\right] & 0 \end{vmatrix} = 0
  1. Determine the order and degree of each of the following differential equations.
    Q.1 i)
    d2ydx2+x(dydx)+y=2sin⁡x\dfrac{d^2y}{dx^2} + x\left(\dfrac{dy}{dx}\right) + y = 2\sin x
  2. Q.1 ii)
    1+(dydx)23=d2ydx2\sqrt[3]{1 + \left(\dfrac{dy}{dx}\right)^2} = \dfrac{d^2y}{dx^2}
  3. Q.1 iii)
    dydx=2sin⁡x+3dydx\dfrac{dy}{dx} = \dfrac{2\sin x + 3}{\dfrac{dy}{dx}}
  4. Q.1 iv)
    d2ydx2+dydx+x=1+d3ydx3\dfrac{d^2y}{dx^2} + \dfrac{dy}{dx} + x = \sqrt{1 + \dfrac{d^3y}{dx^3}}
  5. Q.1 v)
    d2ydt2+(dydt)2+7x+5=0\dfrac{d^2y}{dt^2} + \left(\dfrac{dy}{dt}\right)^2 + 7x + 5 = 0
  6. Q.1 vi)
    (y′′′)2+3y′′+3xy′+5y=0(y''')^2 + 3y'' + 3xy' + 5y = 0
  7. Q.1 vii)
    (d2ydx2)2+cos⁡(dydx)=0\left(\dfrac{d^2y}{dx^2}\right)^2 + \cos\left(\dfrac{dy}{dx}\right) = 0
  8. Q.1 viii)
    [1+(dydx)2]32=8d2ydx2\left[1 + \left(\dfrac{dy}{dx}\right)^2\right]^{\frac{3}{2}} = 8 \dfrac{d^2y}{dx^2}
  9. Q.1 ix)
    (d3ydx3)12−(dydx)13=20\left(\dfrac{d^3y}{dx^3}\right)^{\frac{1}{2}} - \left(\dfrac{dy}{dx}\right)^{\frac{1}{3}} = 20
  10. Q.1 x)
    x+d2ydx2=1+(d2ydx2)2x + \dfrac{d^2y}{dx^2} = \sqrt{1 + \left(\dfrac{d^2y}{dx^2}\right)^2}
  1. Obtain the differential equation by eliminating the arbitrary constants from the following :
    i)
    y=4axy = 4ax
  2. ii)
    y=Ae3x+Be−3xy = Ae^{3x} + Be^{-3x}
  3. iii)
    y=(c1+c2x)exy = (c_1 + c_2 x) e^{x}
  4. iv)
    y=c2+cxy = c^2 + \dfrac{c}{x}
  5. v)
    y=c1e3x+c2e2xy = c_1 e^{3x} + c_2 e^{2x}
  6. 6.3 Ex.2
    The rate of decay of the mass of a radioactive substance any time is kk times its mass at that time, form the differential equation satisfied by the mass of the substance.
  7. 6.3 Ex.3
    Form the differential equation of family of circles above the X-axis and touching the X-axis at the origin.
  8. 6.3 Ex.4
    A particle is moving along the X-axis. Its acceleration at time t is proportional to its velocity at that time. Find the differential equation of the motion of the particle.
  1. Obtain the differential equations by eliminating arbitrary constants
    6.3 Exercise Q.1 i)
    x3+y3=4axx^3 + y^3 = 4ax
  2. 6.3 Exercise Q.1 ii)
    Ax2+By2=1Ax^2 + By^2 = 1
  3. 6.3 Exercise Q.1 iii)
    y=Acos⁡(log⁡x)+Bsin⁡(log⁡x)y = A \cos(\log x) + B \sin(\log x)
  4. 6.3 Exercise Q.1 iv)
    y2=(x+c)3y^2 = (x + c)^3
  5. 6.3 Exercise Q.1 v)
    y=Ae5x+Be−5xy = Ae^{5x} + Be^{-5x}
  6. 6.3 Exercise Q.1 vi)
    (y−a)2=4(x−b)(y - a)^2 = 4(x - b)
  7. 6.3 Exercise Q.1 vii)
    y=a+axy = a + \dfrac{a}{x}
  8. 6.3 Exercise Q.1 viii)
    y=c1e2x+c2e5xy = c_1 e^{2x} + c_2 e^{5x}
  9. 6.3 Exercise Q.1 ix)
    c1x3+c2y2=5c_1 x^3 + c_2 y^2 = 5
  10. 6.3 Exercise Q.1 x)
    y=e−2x(Acos⁡x+Bsin⁡x)y = e^{-2x}(A \cos x + B \sin x)
  11. 6.3 Exercise Q.2
    Form the differential equation of family of lines making intercept a on the X-axis.
  12. 6.3 Exercise Q.3
    Find the differential equation of all parabolas having length of latus rectum 4a4a and axis is parallel to the X-axis.
  13. 6.3 Exercise Q.4
    Find the differential equation of all ellipse whose major axis is twise its minor axis.
  14. 6.3 Exercise Q.5
    Form the differential equation of family of lines parallel to the line 2x+3y+4=02x + 3y + 4 = 0
  15. 6.3 Exercise Q.6
    Find the differential equations of all circles having radius 9 and centre at point A(h,k)A(h, k).
  16. 6.3 Exercise Q.7
    Form the differential equation of all parabolas whose axis is the X-axis.
  1. (verify sec)
    Verify that ysec⁡x=tan⁡x+cy \sec x = \tan x + c is a solution of the differential equation dydx+ytan⁡x=sec⁡x\dfrac{dy}{dx} + y \tan x = \sec x.
  2. (verify log)
    Verify that y=log⁡x+cy = \log x + c is a solution of the differential equation xd2ydx2+dydx=0x \dfrac{d^2 y}{dx^2} + \dfrac{dy}{dx} = 0.
  3. (i) general soln
    Find the general solution of the differential equation dydx=x25−x2\dfrac{dy}{dx} = x \sqrt{25 - x^2}.
  4. (ii) general soln
    Find the general solution of the differential equation dxdt=xlog⁡xt\dfrac{dx}{dt} = \dfrac{x \log x}{t}.
  5. (i) particular soln
    Find the particular solution with given initial conditions: dydx=e2ycos⁡x\dfrac{dy}{dx} = e^{2y} \cos x when x=π6x = \dfrac{\pi}{6}, y=0y = 0.
  6. (ii) particular soln
    Find the particular solution with given initial conditions: y−1y+1+x−1x+1⋅dydx=0\dfrac{y - 1}{y + 1} + \dfrac{x - 1}{x + 1} \cdot \dfrac{dy}{dx} = 0, when x=y=2x = y = 2.
  7. (i) reduce sep
    Reduce the following differential equation to the separated variable form and hence find the general solution: 1+dydx=cosec⁡(x+y)1 + \dfrac{dy}{dx} = \operatorname{cosec}(x + y).
  8. (ii) reduce sep
    Reduce the following differential equation to the separated variable form and hence find the general solution: dydx=(4x+y+1)2\dfrac{dy}{dx} = (4x + y + 1)^2.
  1. In each of the following examples verify that the given expression is a solution of the corresponding differential equation.
    Q.1 (i)
    xy=log⁡y+cxy = \log y + c ; dydx=y21−xy\dfrac{dy}{dx} = \dfrac{y^2}{1 - xy}
  2. Q.1 (ii)
    y=(sin⁡−1x)2+cy = (\sin^{-1} x)^2 + c ; (1−x2)d2ydx2−xdydx=2(1 - x^2) \dfrac{d^2 y}{dx^2} - x \dfrac{dy}{dx} = 2
  3. Q.1 (iii)
    y=e−x+Ax+By = e^{-x} + Ax + B ; exd2ydx2=1e^x \dfrac{d^2 y}{dx^2} = 1
  4. Q.1 (iv)
    y=xmy = x^m ; x2d2ydx2−mxdydx+my=0x^2 \dfrac{d^2 y}{dx^2} - mx \dfrac{dy}{dx} + my = 0
  5. Q.1 (v)
    y=a+bxy = a + \dfrac{b}{x} ; xd2ydx2+2dydx=0x \dfrac{d^2 y}{dx^2} + 2 \dfrac{dy}{dx} = 0
  6. Q.1 (vi)
    y=eaxy = e^{ax} ; xdydx=ylog⁡yx \dfrac{dy}{dx} = y \log y
  7. Solve the following differential equations.
    Q.2 (i)
    dydx=1+y21+x2\dfrac{dy}{dx} = \dfrac{1 + y^2}{1 + x^2}
  8. Q.2 (ii)
    log⁡(dydx)=2x+3y\log\left(\dfrac{dy}{dx}\right) = 2x + 3y
  9. Q.2 (iii)
    y−xdydx=0y - x \dfrac{dy}{dx} = 0
  10. Q.2 (iv)
    sec⁡2x⋅tan⁡y⋅dx+sec⁡2y⋅tan⁡x⋅dy=0\sec^2 x \cdot \tan y \cdot dx + \sec^2 y \cdot \tan x \cdot dy = 0
  11. Q.2 (v)
    cos⁡x⋅cos⁡y⋅dy−sin⁡x⋅sin⁡y⋅dx=0\cos x \cdot \cos y \cdot dy - \sin x \cdot \sin y \cdot dx = 0
  12. Q.2 (vi)
    dydx=−k\dfrac{dy}{dx} = -k, where kk = constant.
  13. Q.2 (vii)
    cos⁡2y⋅dyx+cos⁡2x⋅dxy=0\dfrac{\cos^2 y \cdot dy}{x} + \dfrac{\cos^2 x \cdot dx}{y} = 0
  14. Q.2 (viii)
    y3−dydx=x2dydxy^3 - \dfrac{dy}{dx} = x^2 \dfrac{dy}{dx}
  15. Q.2 (ix)
    2ex+2y⋅dx−3dy=02 e^{x + 2y} \cdot dx - 3 dy = 0
  16. Q.2 (x)
    dydx=ex+y+x2ey\dfrac{dy}{dx} = e^{x + y} + x^2 e^y
  17. For each of the following differential equations find the particular solution satisfying the given condition.
    Q.3 (i)
    3extan⁡y⋅dx+(1+ex)sec⁡2y⋅dy=03 e^x \tan y \cdot dx + (1 + e^x) \sec^2 y \cdot dy = 0, when x=0x = 0, y=πy = \pi.
  18. Q.3 (ii)
    (x−y2x)⋅dx−(y+x2y)⋅dy=0(x - y^2 x) \cdot dx - (y + x^2 y) \cdot dy = 0, when x=2x = 2, y=0y = 0.
  19. Q.3 (iii)
    y(1+log⁡x)dxdy−xlog⁡x=0y (1 + \log x) \dfrac{dx}{dy} - x \log x = 0, y=e2y = e^2, when x=ex = e.
  20. Q.3 (iv)
    (ey+1)cos⁡x+eysin⁡xdydx=0(e^y + 1) \cos x + e^y \sin x \dfrac{dy}{dx} = 0, when x=π6x = \dfrac{\pi}{6}, y=0y = 0.
  21. Q.3 (v)
    (x+1)dydx−1=2e−y(x + 1) \dfrac{dy}{dx} - 1 = 2 e^{-y}, y=0y = 0, x=1x = 1.
  22. Q.3 (vi)
    cos⁡(dydx)=a\cos\left(\dfrac{dy}{dx}\right) = a, a∈Ra \in \mathbb{R}, y(0)=2y(0) = 2.
  23. Reduce each of the following differential to the variable separable form and hence solve.
    Q.4 (i)
    dydx=cos⁡(x+y)\dfrac{dy}{dx} = \cos(x + y)
  24. Q.4 (ii)
    (x−y)2dydx=a2(x - y)^2 \dfrac{dy}{dx} = a^2
  25. Q.4 (iii)
    x+ydydx=sec⁡(x2+y2)x + y \dfrac{dy}{dx} = \sec(x^2 + y^2)
  26. Q.4 (iv)
    cos⁡2(x−2y)=1−2dydx\cos^2(x - 2y) = 1 - 2 \dfrac{dy}{dx}
  27. Q.4 (v)
    (2x−2y+3)dx−(x−y+1)dy=0(2x - 2y + 3) dx - (x - y + 1) dy = 0, when x=0x = 0, y=1y = 1.
  1. (i) homogeneous
    Solve the differential equation x2y⋅dx−(x3+y3)⋅dy=0x^2 y \cdot dx - (x^3 + y^3) \cdot dy = 0.
  2. (ii) homogeneous
    Solve the differential equation xdydx=xtan⁡(yx)+yx \dfrac{dy}{dx} = x \tan\left(\dfrac{y}{x}\right) + y.
  3. (iii) homogeneous
    Solve the differential equation dydx=y+x2+y2x\dfrac{dy}{dx} = \dfrac{y + \sqrt{x^2 + y^2}}{x}.
  1. Solve the following differential equations:
    Q.1
    xsin⁡(yx)dy=[ysin⁡(yx)−x]dxx \sin\left(\dfrac{y}{x}\right) dy = \left[y \sin\left(\dfrac{y}{x}\right) - x\right] dx
  2. Q.2
    (x2−y2)dx−2xy⋅dy=0(x^2 - y^2) dx - 2xy \cdot dy = 0
  3. Q.3
    (1+2exy)+2exy(1−xy)dydx=0\left(1 + 2 e^{\frac{x}{y}}\right) + 2 e^{\frac{x}{y}}\left(1 - \dfrac{x}{y}\right) \dfrac{dy}{dx} = 0
  4. Q.4
    y2⋅dx+(xy+x2)dy=0y^2 \cdot dx + (xy + x^2) dy = 0
  5. Q.5
    (x2−y2)dx+2xy⋅dy=0(x^2 - y^2) dx + 2xy \cdot dy = 0
  6. Q.6
    dydx+x−2y2x−y=0\dfrac{dy}{dx} + \dfrac{x - 2y}{2x - y} = 0
  7. Q.7
    xdydx−y+xsin⁡(yx)=0x \dfrac{dy}{dx} - y + x \sin\left(\dfrac{y}{x}\right) = 0
  8. Q.8
    (1+exy)dx+exy(1−xy)dy=0\left(1 + e^{\frac{x}{y}}\right) dx + e^{\frac{x}{y}}\left(1 - \dfrac{x}{y}\right) dy = 0
  9. Q.9
    y2−x2dydx=xydydxy^2 - x^2 \dfrac{dy}{dx} = xy \dfrac{dy}{dx}
  10. Q.10
    xydydx=x2+2y2xy \dfrac{dy}{dx} = x^2 + 2y^2, y(1)=0y(1) = 0
  11. Q.11
    x dy+2y⋅dx=0x \, dy + 2y \cdot dx = 0, when x=2x = 2, y=1y = 1
  12. Q.12
    x2dydx=x2+xy+y2x^2 \dfrac{dy}{dx} = x^2 + xy + y^2
  13. Q.13
    (9x+5y)dy+(15x+11y)dx=0(9x + 5y) dy + (15x + 11y) dx = 0
  14. Q.14
    (x2+3xy+y2)dx−x2 dy=0(x^2 + 3xy + y^2) dx - x^2 \, dy = 0
  15. Q.15
    (x2+y2)dx−2xy⋅dy=0(x^2 + y^2) dx - 2xy \cdot dy = 0
  1. (i) linear
    Solve the differential equation dydx+y=e−x\dfrac{dy}{dx} + y = e^{-x}.
  2. (ii) linear
    Solve the differential equation xsin⁡xdydx+(xcos⁡x+sin⁡x)y=sin⁡xx \sin x \dfrac{dy}{dx} + (x \cos x + \sin x) y = \sin x.
  3. (iii) linear
    Solve the differential equation (1+y2)dx=(tan⁡−1y−x)dy(1 + y^2) dx = (\tan^{-1} y - x) dy.
  4. (slope of tangent)
    The slope of the tangent to the curve at any point is equal to y+2xy + 2x. Find the equation of the curve passing through the origin.
  1. Solve the following differential equations:
    Q.1 (i)
    dydx+yx=x3−3\dfrac{dy}{dx} + \dfrac{y}{x} = x^3 - 3
  2. Q.1 (ii)
    cos⁡2xdydx+y=tan⁡x\cos^2 x \dfrac{dy}{dx} + y = \tan x
  3. Q.1 (iii)
    (x+2y3)dydx=y(x + 2y^3) \dfrac{dy}{dx} = y
  4. Q.1 (iv)
    dydx+ysec⁡x=tan⁡x\dfrac{dy}{dx} + y \sec x = \tan x
  5. Q.1 (v)
    xdydx+2y=x2log⁡xx \dfrac{dy}{dx} + 2y = x^2 \log x
  6. Q.1 (vi)
    (x+y)dydx=1(x + y) \dfrac{dy}{dx} = 1
  7. Q.1 (vii)
    (x+a)dydx−3y=(x+a)5(x + a) \dfrac{dy}{dx} - 3y = (x + a)^5
  8. Q.1 (viii)
    dr+(2rcot⁡θ+sin⁡2θ)dθ=0dr + (2r \cot \theta + \sin 2\theta) d\theta = 0
  1. 6.5 Ex.1
    The population of a town increasing at a rate proportional to the population at that time. If the population increases from 40 thousands to 60 thousands in 40 years, what will be the population in another 20 years. (Given 32=1.2247)\left(\text{Given } \sqrt{\dfrac{3}{2}} = 1.2247\right).
  2. 6.5 Ex.2
    Bacteria increase at the rate proportional to the number of bacteria present. If the original number NN doubles in 3 hours, find in how many hours the number of bacteria will be 4N4N?
  3. 6.5 Ex.3
    Bismath has half life of 5 days. A sample originally has a mass of 800 mg. Find the mass remaining after 30 days.
  4. 6.5 Ex.4
    Water at 100∘100^{\circ}c cools in 10 minutes to 88∘88^{\circ}c in a room temperature of 25∘25^{\circ}c. Find the temperature of water after 20 minutes.
  5. 6.5 Ex.5
    Water is being poured into a vessel in the form of an inverted right circular cone of semi vertical angle 45∘45^{\circ}c in such a way that the rate of change of volume at any moment is proportional to the area of the curved surfaces which is wet at that moment. Initially, the vessel is full to a height of 2 cms. And after 2 seconds the height becomes 10 cm. Show that after 3.5 seconds from that start, the height of water will be 16 cms.
  1. Q.1
    In a certain culture of bacteria the rate of increase is proportional to the number present. If it is found that the number doubles in 4 hours, find the number of times the bacteria are increased in 12 hours.
  2. Q.2
    If the population of a country doubles in 60 years, in how many years will it be triple (treble) under the assumption that the rate of increase is proportional to the number of inhabitants? [Given log⁡2=0.6912\log 2 = 0.6912, log⁡3=1.0986\log 3 = 1.0986]
  3. Q.3
    If a body cools from 80∘80^{\circ}c to 50∘50^{\circ}c at room temperature of 25∘25^{\circ}c in 30 minutes, find the temperature of the body after 1 hour.
  4. Q.4
    The rate of growth of bacteria is proportional to the number present. If initially, there were 1000 bacteria and the number double in 1 hour, find the number of bacteria after 2122\dfrac{1}{2} hours. [Take 2=1.414\sqrt{2} = 1.414]
  5. Q.5
    The rate of disintegration of a radio active element at any time tt is proportional to its mass at that time. Find the time during which the original mass of 1.51.5 gm. will disintegrate into its mass of 0.50.5 gm.
  6. Q.6
    The rate of decay of certain substance is directly proportional to the amount present at that instant. Initially, there are 25 gms of certain substance and two hours later it is found that 9 gms are left. Find the amount left after one more hour.
  7. Q.7
    Find the population of a city at any time tt, given that the rate of increase of population is proportional to the population at the instant and that in a period of 40 years the population increased from 30,000 to 40,000.
  8. Q.8
    A body cools according to Newton's law from 100∘100^{\circ}c to 60∘60^{\circ}c in 20 minutes. The temperature of the surrounding being 20∘20^{\circ}c how long will it take to cool down to 30∘30^{\circ}c?
  9. Q.9
    A right circular cone has height 9 cms and radius of the base 5 cms. It is inverted and water is poured into it. If at any instant the water level rises at the rate of πA\dfrac{\pi}{A} cms/sec. where AA is the area of water surface at that instant, show that the vessel will the full in 75 seconds.
  10. Q.10
    Assume that a spherical raindrop evaporates at a rate proportional to its surface area. If its radius originally is 3mm and 1 hour later has been reduced to 2mm, find an expression for the radius of the raindrop at any time tt.
  11. Q.11
    The rate of growth of the population of a city at any time tt is proportional to the size of the population. For a certain city it is found that the constant of proportionality is 0.04. Find the population of the city after 25 years if the initial population is 10,000. [Take e=2.7182e = 2.7182]
  12. Q.12
    Radium decomposes at the rate proportional to the amount present at any time. If pp percent of amount disappears in one year, what percent of amount of radium will be left after 2 years?
  1. Misc I (1)
    The order and degree of the differential equation 1+(dydx)2=(d2ydx2)32\sqrt{1 + \left(\dfrac{dy}{dx}\right)^{2}} = \left(\dfrac{d^{2}y}{dx^{2}}\right)^{\frac{3}{2}} are respectively ...
    1. A.
      2,12, 1
    2. B.
      1,21, 2
    3. C.
      3,23, 2
    4. D.
      2,32, 3
  2. Misc I (2)
    The differential equation of y=c2+cxy = c^{2} + \dfrac{c}{x} is ...
    1. A.
      x4(dydx)2−xdydx=yx^{4}\left(\dfrac{dy}{dx}\right)^{2} - x\dfrac{dy}{dx} = y
    2. B.
      d2ydx2+xdydx+y=0\dfrac{d^{2}y}{dx^{2}} + x\dfrac{dy}{dx} + y = 0
    3. C.
      x3(dydx)2+xdydx=yx^{3}\left(\dfrac{dy}{dx}\right)^{2} + x\dfrac{dy}{dx} = y
    4. D.
      d2ydx2+dydx−y=0\dfrac{d^{2}y}{dx^{2}} + \dfrac{dy}{dx} - y = 0
  3. Misc I (3)
    x2+y2=a2x^{2} + y^{2} = a^{2} is a solution of ...
    1. A.
      d2ydx2+dydx−y=0\dfrac{d^{2}y}{dx^{2}} + \dfrac{dy}{dx} - y = 0
    2. B.
      y=x1+(dydx)2+a2yy = x\sqrt{1 + \left(\dfrac{dy}{dx}\right)^{2}} + a^{2}y
    3. C.
      y=xdydx+a1+(dydx)2y = x\dfrac{dy}{dx} + a\sqrt{1 + \left(\dfrac{dy}{dx}\right)^{2}}
    4. D.
      d2ydx2=(x+1)dydx\dfrac{d^{2}y}{dx^{2}} = (x + 1)\dfrac{dy}{dx}
  4. Misc I (4)
    The differential equation of all circles having their centers on the line y=5y = 5 and touching the X-axis is
    1. A.
      y2(1+dydx)=25y^{2}\left(1 + \dfrac{dy}{dx}\right) = 25
    2. B.
      (y−5)2[1+(dydx)2]=25(y - 5)^{2}\left[1 + \left(\dfrac{dy}{dx}\right)^{2}\right] = 25
    3. C.
      (y−5)2+[1+(dydx)2]=25(y - 5)^{2} + \left[1 + \left(\dfrac{dy}{dx}\right)^{2}\right] = 25
    4. D.
      (y−5)2[1−(dydx)2]=25(y - 5)^{2}\left[1 - \left(\dfrac{dy}{dx}\right)^{2}\right] = 25
  5. Misc I (5)
    The differential equation ydydx+x=0y\dfrac{dy}{dx} + x = 0 represents family of ...
    1. A.
      circles
    2. B.
      parabolas
    3. C.
      ellipses
    4. D.
      hyper bolas
  6. Misc I (6)
    The solution of 1x⋅dydx=tan⁡−1x\dfrac{1}{x} \cdot \dfrac{dy}{dx} = \tan^{-1} x is ...
    1. A.
      x2tan⁡−1x2+c=0\dfrac{x^{2}\tan^{-1} x}{2} + c = 0
    2. B.
      xtan⁡−1x+c=0x\tan^{-1} x + c = 0
    3. C.
      x−tan⁡−1x=cx - \tan^{-1} x = c
    4. D.
      y=x2tan⁡−1x2−12(x−tan⁡−1x)+cy = \dfrac{x^{2}\tan^{-1} x}{2} - \dfrac{1}{2}(x - \tan^{-1} x) + c
  7. Misc I (7)
    The solution of (x+y)2dydx=1(x + y)^{2}\dfrac{dy}{dx} = 1 is ...
    1. A.
      x=tan⁡−1(x+y)+cx = \tan^{-1}(x + y) + c
    2. B.
      ytan⁡−1(xy)=cy\tan^{-1}\left(\dfrac{x}{y}\right) = c
    3. C.
      y=tan⁡−1(x+y)+cy = \tan^{-1}(x + y) + c
    4. D.
      y+tan⁡−1(x+y)=cy + \tan^{-1}(x + y) = c
  8. Misc I (8)
    The solution of dydx=y+x2−y22\dfrac{dy}{dx} = \dfrac{y + \sqrt{x^{2} - y^{2}}}{2} is ...
    1. A.
      sin⁡−1(yx)=2log⁡∣x∣+c\sin^{-1}\left(\dfrac{y}{x}\right) = 2\log|x| + c
    2. B.
      sin⁡−1(yx)=log⁡∣x∣+c\sin^{-1}\left(\dfrac{y}{x}\right) = \log|x| + c
    3. C.
      sin⁡(xy)=log⁡∣x∣+c\sin\left(\dfrac{x}{y}\right) = \log|x| + c
    4. D.
      sin⁡(yx)=log⁡∣y∣+c\sin\left(\dfrac{y}{x}\right) = \log|y| + c
  9. Misc I (9)
    The solution of dydx+y=cos⁡x−sin⁡x\dfrac{dy}{dx} + y = \cos x - \sin x is ...
    1. A.
      y ex=cos⁡x+cy\, e^{x} = \cos x + c
    2. B.
      y ex+excos⁡x=cy\, e^{x} + e^{x}\cos x = c
    3. C.
      y ex=excos⁡x+cy\, e^{x} = e^{x}\cos x + c
    4. D.
      y2 ex=excos⁡x+cy^{2}\, e^{x} = e^{x}\cos x + c
  10. Misc I (10)
    The integrating factor of linear differential equation xdydx+2y=x2log⁡xx\dfrac{dy}{dx} + 2y = x^{2}\log x is ...
    1. A.
      1x\dfrac{1}{x}
    2. B.
      kk
    3. C.
      1n2\dfrac{1}{n^{2}}
    4. D.
      x2x^{2}
  11. Misc I (11)
    The solution of the differential equation dydx=sec⁡x−ytan⁡x\dfrac{dy}{dx} = \sec x - y\tan x is
    1. A.
      ysec⁡x+tan⁡x=cy\sec x + \tan x = c
    2. B.
      ysec⁡x=tan⁡x+cy\sec x = \tan x + c
    3. C.
      sec⁡x+ytan⁡x=c\sec x + y\tan x = c
    4. D.
      sec⁡x=ytan⁡x+c\sec x = y\tan x + c
  12. Misc I (12)
    The particular solution of dydx=xey−x\dfrac{dy}{dx} = xe^{y - x}, when x=y=0x = y = 0 is ...
    1. A.
      ex−y=x+1e^{x - y} = x + 1
    2. B.
      ex+y=x+1e^{x + y} = x + 1
    3. C.
      ex+ey=x+1e^{x} + e^{y} = x + 1
    4. D.
      ey−x=x−1e^{y - x} = x - 1
  13. Misc I (13)
    x2a2−y2b2=1\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1 is a solution of ...
    1. A.
      d2ydx2+yx+(dydx)2=0\dfrac{d^{2}y}{dx^{2}} + yx + \left(\dfrac{dy}{dx}\right)^{2} = 0
    2. B.
      xyd2ydx2+2(dydx)2−ydydx=0xy\dfrac{d^{2}y}{dx^{2}} + 2\left(\dfrac{dy}{dx}\right)^{2} - y\dfrac{dy}{dx} = 0
    3. C.
      yd2ydx2+2(dydx)2+y=0y\dfrac{d^{2}y}{dx^{2}} + 2\left(\dfrac{dy}{dx}\right)^{2} + y = 0
    4. D.
      xydydx+yd2ydx2=0xy\dfrac{dy}{dx} + y\dfrac{d^{2}y}{dx^{2}} = 0
  14. Misc I (14)
    The decay rate of certain substance is directly proporational to the amount present at that instant. Initially there are 27 grams of substance and 3 hours later it is found that 8 grams left. The amount left after one more hour is...
    1. A.
      5235\dfrac{2}{3} grams
    2. B.
      5135\dfrac{1}{3} grams
    3. C.
      5⋅15 \cdot 1 grams
    4. D.
      55 grams
  15. Misc I (15)
    If the surrounding air is kept at 20∘20^{\circ}c and a body cools from 80∘80^{\circ}c to 70∘70^{\circ}c in 5 minutes, the temparature of the body after 15 minutes will be...
    1. A.
      51⋅7∘51 \cdot 7^{\circ}c
    2. B.
      54⋅7∘54 \cdot 7^{\circ}c
    3. C.
      52⋅7∘52 \cdot 7^{\circ}c
    4. D.
      50⋅7∘50 \cdot 7^{\circ}c
  1. Determine the order and degree of the following differential equations :
    Misc II Q.1 i)
    d2ydx2+5dydx+y=x3\dfrac{d^{2}y}{dx^{2}} + 5\dfrac{dy}{dx} + y = x^{3}
  2. Misc II Q.1 ii)
    (d3ydx3)2=1+dydx5\left(\dfrac{d^{3}y}{dx^{3}}\right)^{2} = \sqrt[5]{1 + \dfrac{dy}{dx}}
  3. Misc II Q.1 iii)
    1+(dydx)23=d2ydx2\sqrt[3]{1 + \left(\dfrac{dy}{dx}\right)^{2}} = \dfrac{d^{2}y}{dx^{2}}
  4. Misc II Q.1 iv)
    dydx=3y+1+5(dydx)24\dfrac{dy}{dx} = 3y + \sqrt[4]{1 + 5\left(\dfrac{dy}{dx}\right)^{2}}
  5. Misc II Q.1 v)
    d4ydx4+sin⁡(dydx)=0\dfrac{d^{4}y}{dx^{4}} + \sin\left(\dfrac{dy}{dx}\right) = 0
  6. In each of the following examples, verify that the given function is a solution of the differential equation.
    Misc II Q.2 i)
    x2+y2=r2, xdydx+r1+(dydx)2=yx^{2} + y^{2} = r^{2},\ x\dfrac{dy}{dx} + r\sqrt{1 + \left(\dfrac{dy}{dx}\right)^{2}} = y
  7. Misc II Q.2 ii)
    y=eaxsin⁡bx, d2ydx2−2adydx+(a2+b2) y=0y = e^{ax}\sin bx,\ \dfrac{d^{2}y}{dx^{2}} - 2a\dfrac{dy}{dx} + (a^{2} + b^{2})\, y = 0
  8. Misc II Q.2 iii)
    y=3cos⁡(log⁡x)+4sin⁡(log⁡x), xd2ydx2+xdydx+y=0y = 3\cos(\log x) + 4\sin(\log x),\ x\dfrac{d^{2}y}{dx^{2}} + x\dfrac{dy}{dx} + y = 0
  9. Misc II Q.2 iv)
    y=aex+be−x+x2, xd2ydx2+2dydx+x3=xy+2y = ae^{x} + be^{-x} + x^{2},\ x\dfrac{d^{2}y}{dx^{2}} + 2\dfrac{dy}{dx} + x^{3} = xy + 2
  10. Misc II Q.2 v)
    x2=2y2log⁡y, x2+y2=xydxdyx^{2} = 2y^{2}\log y,\ x^{2} + y^{2} = xy\dfrac{dx}{dy}
  11. Obtain the differential equation by eliminating the arbitrary constants from the following equations.
    Misc II Q.3 i)
    y2=a(b−x)(b+x)y^{2} = a(b - x)(b + x)
  12. Misc II Q.3 ii)
    y=asin⁡(x+b)y = a\sin(x + b)
  13. Misc II Q.3 iii)
    (y−a)2=b(x+4)(y - a)^{2} = b(x + 4)
  14. Misc II Q.3 iv)
    y=acos⁡(log⁡x)+bsin⁡(log⁡x)y = \sqrt{a}\cos(\log x) + b\sin(\log x)
  15. Misc II Q.3 v)
    y=Ae3x+1+Be−3x+1y = Ae^{3x + 1} + Be^{-3x + 1}
  16. Form the differential equation of :
    Misc II Q.4 i)
    all circles which pass through the origin and whose centres lie on X-axis.
  17. Misc II Q.4 ii)
    all parabolas which have 4b4b as latus rectum and whose axes is parallel to Y-axis.
  18. Misc II Q.4 iii)
    all ellipse whose major axis is twice its minor axis.
  19. Misc II Q.4 iv)
    all the lines which are parallel to the line 3x−2y+7=03x - 2y + 7 = 0.
  20. Misc II Q.4 v)
    the hyperbola whose length of transverse and conjugate axes are half of that of the given hyperbola x216−y236=k\dfrac{x^{2}}{16} - \dfrac{y^{2}}{36} = k.
  21. Solve the following differential equations :
    Misc II Q.5 i)
    log⁡(dydx)=2x+3y\log\left(\dfrac{dy}{dx}\right) = 2x + 3y
  22. Misc II Q.5 ii)
    dydx=x2y+y\dfrac{dy}{dx} = x^{2}y + y
  23. Misc II Q.5 iii)
    dydx=2y−x2y+x\dfrac{dy}{dx} = \dfrac{2y - x}{2y + x}
  24. Misc II Q.5 iv)
    x dy=(x+y+1) dxx\, dy = (x + y + 1)\, dx
  25. Misc II Q.5 v)
    dydx+ycot⁡x=x2cot⁡x+2x\dfrac{dy}{dx} + y\cot x = x^{2}\cot x + 2x
  26. Misc II Q.5 vi)
    ylog⁡y=(log⁡y2−x)dydxy\log y = (\log y^{2} - x)\dfrac{dy}{dx}
  27. Misc II Q.5 vii)
    4dxdy+8x=5e−3y4\dfrac{dx}{dy} + 8x = 5e^{-3y}
  28. Find the particular solution of the following differential equations :
    Misc II Q.6 (1)
    y(1+log⁡x)=(log⁡xx)dydxy(1 + \log x) = (\log x^{x})\dfrac{dy}{dx}, when y(e)=e2y(e) = e^{2}
  29. Misc II Q.6 (2)
    (x+2y2)dydx=y(x + 2y^{2})\dfrac{dy}{dx} = y, when x=2,y=1x = 2, y = 1
  30. Misc II Q.6 (3)
    dydx−3ycot⁡x=sin⁡2x\dfrac{dy}{dx} - 3y\cot x = \sin 2x, when y(π2)=2y\left(\dfrac{\pi}{2}\right) = 2
  31. Misc II Q.6 (4)
    (x+y) dy+(x−y) dx=0(x + y)\, dy + (x - y)\, dx = 0, when x=1=yx = 1 = y
  32. Misc II Q.6 (5)
    2exy dx+(y−2xexy)dy=02e^{\frac{x}{y}}\, dx + \left(y - 2xe^{\frac{x}{y}}\right)dy = 0, when y(0)=1y(0) = 1
  33. Misc II Q.7
    Show that the general solution of the differential equation dydx=y2+y+1x2+x+1\dfrac{dy}{dx} = \dfrac{y^{2} + y + 1}{x^{2} + x + 1} is given by (x+y+1)=c(1−x−y−2xy)(x + y + 1) = c(1 - x - y - 2xy)
  34. Misc II Q.8
    The normal lines to a given curve at each point (x,y)(x, y) on the curve pass through (2,0)(2, 0). The curve passes through (2,3)(2, 3). Find the equation of the curve.
  35. Misc II Q.9
    The volume of spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 units. Find the radius of the balloon after tt second.
  36. Misc II Q.10
    A person's assets start reducing in such a way that the rate of reduction of assets is proportional to the square root of the assets existing at that moment. If the assets at the begining are Rs. 10 lakhs and they dwindle down to Rs. 10,000 after 2 years, show that the person will be bankrupt in 2292\dfrac{2}{9} years from the start.