Mathematics · Textbook solutions

Differential Equations

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 127 questions

9.2 Basic Concepts — Order and Degree

15 q

Solved Examples

Worked · 3
  1. Find the order and degree, if defined, of each of the following differential equations:
    9.1 Eg.1(i)
    dydx−cos⁡x=0\frac{dy}{dx} - \cos x = 0
  2. 9.1 Eg.1(ii)
    xyd2ydx2+x(dydx)2−ydydx=0xy\frac{d^{2}y}{dx^{2}} + x\left(\frac{dy}{dx}\right)^{2} - y\frac{dy}{dx} = 0
  3. 9.1 Eg.1(iii)
    y′′′+y2+ey′=0y''' + y^{2} + e^{y'} = 0

Exercise 9.1

Practice · 12
  1. Ex 9.1 Q1
    Determine order and degree (if defined) of the differential equation d4ydx4+sin⁡(y′′′)=0\frac{d^{4}y}{dx^{4}} + \sin(y''') = 0.
  2. Ex 9.1 Q2
    Determine order and degree (if defined) of the differential equation y′+5y=0y' + 5y = 0.
  3. Ex 9.1 Q3
    Determine order and degree (if defined) of the differential equation (dsdt)4+3sd2sdt2=0\left(\frac{ds}{dt}\right)^{4} + 3s\frac{d^{2}s}{dt^{2}} = 0.
  4. Ex 9.1 Q4
    Determine order and degree (if defined) of the differential equation (d2ydx2)2+cos⁡(dydx)=0\left(\frac{d^{2}y}{dx^{2}}\right)^{2} + \cos\left(\frac{dy}{dx}\right) = 0.
  5. Ex 9.1 Q5
    Determine order and degree (if defined) of the differential equation d2ydx2=cos⁡3x+sin⁡3x\frac{d^{2}y}{dx^{2}} = \cos 3x + \sin 3x.
  6. Ex 9.1 Q6
    Determine order and degree (if defined) of the differential equation (y′′′)2+(y′′)3+(y′)4+y5=0(y''')^{2} + (y'')^{3} + (y')^{4} + y^{5} = 0.
  7. Ex 9.1 Q7
    Determine order and degree (if defined) of the differential equation y′′′+2y′′+y′=0y''' + 2y'' + y' = 0.
  8. Ex 9.1 Q8
    Determine order and degree (if defined) of the differential equation y′+y=exy' + y = e^{x}.
  9. Ex 9.1 Q9
    Determine order and degree (if defined) of the differential equation y′′+(y′)2+2y=0y'' + (y')^{2} + 2y = 0.
  10. Ex 9.1 Q10
    Determine order and degree (if defined) of the differential equation y′′+2y′+sin⁡y=0y'' + 2y' + \sin y = 0.
  11. Ex 9.1 Q11
    The degree of the differential equation (d2ydx2)3+(dydx)2+sin⁡(dydx)+1=0\left(\frac{d^{2}y}{dx^{2}}\right)^{3} + \left(\frac{dy}{dx}\right)^{2} + \sin\left(\frac{dy}{dx}\right) + 1 = 0 is
    1. A.
      3
    2. B.
      2
    3. C.
      1
    4. D.
      not defined
  12. Ex 9.1 Q12
    The order of the differential equation 2x2d2ydx2−3dydx+y=02x^{2}\frac{d^{2}y}{dx^{2}} - 3\frac{dy}{dx} + y = 0 is
    1. A.
      2
    2. B.
      1
    3. C.
      0
    4. D.
      not defined

9.3 General and Particular Solutions of a Differential Equation

14 q

Solved Examples

Worked · 2
  1. 9.2 Eg.2
    Verify that the function y=e−3xy = e^{-3x} is a solution of the differential equation d2ydx2+dydx−6y=0\frac{d^2y}{dx^2} + \frac{dy}{dx} - 6y = 0.
  2. 9.2 Eg.3
    Verify that the function y=acos⁡x+bsin⁡xy = a\cos x + b\sin x, where a,b∈Ra, b \in \mathbf{R} is a solution of the differential equation d2ydx2+y=0\frac{d^2y}{dx^2} + y = 0.

Exercise 9.2

Practice · 12
  1. Ex 9.2 Q1
    Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation: y=ex+1y = e^x + 1 : y′′−y′=0y'' - y' = 0
  2. Ex 9.2 Q2
    Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation: y=x2+2x+Cy = x^2 + 2x + \mathrm{C} : y′−2x−2=0y' - 2x - 2 = 0
  3. Ex 9.2 Q3
    Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation: y=cos⁡x+Cy = \cos x + \mathrm{C} : y′+sin⁡x=0y' + \sin x = 0
  4. Ex 9.2 Q4
    Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation: y=1+x2y = \sqrt{1 + x^2} : y′=xy1+x2y' = \frac{xy}{1 + x^2}
  5. Ex 9.2 Q5
    Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation: y=Axy = \mathrm{A}x : xy′=y (x≠0)xy' = y\ (x \neq 0)
  6. Ex 9.2 Q6
    Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation: y=xsin⁡xy = x\sin x : xy′=y+xx2−y2xy' = y + x\sqrt{x^2 - y^2} (x≠0x \neq 0 and x>yx > y or x<−yx < -y)
  7. Ex 9.2 Q7
    Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation: xy=log⁡y+Cxy = \log y + \mathrm{C} : y′=y21−xy (xy≠1)y' = \frac{y^2}{1 - xy}\ (xy \neq 1)
  8. Ex 9.2 Q8
    Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation: y−cos⁡y=xy - \cos y = x : (ysin⁡y+cos⁡y+x) y′=y(y\sin y + \cos y + x)\,y' = y
  9. Ex 9.2 Q9
    Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation: x+y=tan⁡−1yx + y = \tan^{-1} y : y2y′+y2+1=0y^2 y' + y^2 + 1 = 0
  10. Ex 9.2 Q10
    Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation: y=a2−x2y = \sqrt{a^2 - x^2}, x∈(−a,a)x \in (-a, a) : x+y dydx=0 (y≠0)x + y\,\frac{dy}{dx} = 0\ (y \neq 0)
  11. Ex 9.2 Q11
    The number of arbitrary constants in the general solution of a differential equation of fourth order are:
    1. A.
      0
    2. B.
      2
    3. C.
      3
    4. D.
      4
  12. Ex 9.2 Q12
    The number of arbitrary constants in the particular solution of a differential equation of third order are:
    1. A.
      3
    2. B.
      2
    3. C.
      1
    4. D.
      0

9.4.1 Differential Equations with Variables Separable

29 q

Solved Examples

Worked · 6
  1. 9.3 Eg.4
    Find the general solution of the differential equation dydx=x+12−y\frac{dy}{dx} = \frac{x+1}{2-y}, (y≠2)(y \neq 2).
  2. 9.3 Eg.5
    Find the general solution of the differential equation dydx=1+y21+x2\frac{dy}{dx} = \frac{1+y^2}{1+x^2}.
  3. 9.3 Eg.6
    Find the particular solution of the differential equation dydx=−4xy2\frac{dy}{dx} = -4xy^2 given that y=1y = 1, when x=0x = 0.
  4. 9.3 Eg.7
    Find the equation of the curve passing through the point (1,1)(1, 1) whose differential equation is x dy=(2x2+1) dxx\,dy = (2x^2 + 1)\,dx (x≠0)(x \neq 0).
  5. 9.3 Eg.8
    Find the equation of a curve passing through the point (−2,3)(-2, 3), given that the slope of the tangent to the curve at any point (x,y)(x, y) is 2xy2\frac{2x}{y^2}.
  6. 9.3 Eg.9
    In a bank, principal increases continuously at the rate of 5% per year. In how many years Rs 1000 double itself?

Exercise 9.3

Practice · 23
  1. Ex 9.3 Q1
    Find the general solution of the differential equation: dydx=1−cos⁡x1+cos⁡x\frac{dy}{dx} = \frac{1 - \cos x}{1 + \cos x}
  2. Ex 9.3 Q2
    Find the general solution of the differential equation: dydx=4−y2 (−2<y<2)\frac{dy}{dx} = \sqrt{4 - y^2}\ (-2 < y < 2)
  3. Ex 9.3 Q3
    Find the general solution of the differential equation: dydx+y=1 (y≠1)\frac{dy}{dx} + y = 1\ (y \neq 1)
  4. Ex 9.3 Q4
    Find the general solution of the differential equation: sec⁡2xtan⁡y dx+sec⁡2ytan⁡x dy=0\sec^2 x \tan y\,dx + \sec^2 y \tan x\,dy = 0
  5. Ex 9.3 Q5
    Find the general solution of the differential equation: (ex+e−x) dy−(ex−e−x) dx=0(e^x + e^{-x})\,dy - (e^x - e^{-x})\,dx = 0
  6. Ex 9.3 Q6
    Find the general solution of the differential equation: dydx=(1+x2)(1+y2)\frac{dy}{dx} = (1 + x^2)(1 + y^2)
  7. Ex 9.3 Q7
    Find the general solution of the differential equation: ylog⁡y dx−x dy=0y \log y\,dx - x\,dy = 0
  8. Ex 9.3 Q8
    Find the general solution of the differential equation: x5dydx=−y5x^5 \frac{dy}{dx} = -y^5
  9. Ex 9.3 Q9
    Find the general solution of the differential equation: dydx=sin⁡−1x\frac{dy}{dx} = \sin^{-1} x
  10. Ex 9.3 Q10
    Find the general solution of the differential equation: extan⁡y dx+(1−ex)sec⁡2y dy=0e^x \tan y\,dx + (1 - e^x)\sec^2 y\,dy = 0
  11. Ex 9.3 Q11
    Find a particular solution of the differential equation satisfying the given condition: (x3+x2+x+1)dydx=2x2+x(x^3 + x^2 + x + 1)\frac{dy}{dx} = 2x^2 + x; y=1y = 1 when x=0x = 0
  12. Ex 9.3 Q12
    Find a particular solution of the differential equation satisfying the given condition: x(x2−1)dydx=1x(x^2 - 1)\frac{dy}{dx} = 1; y=0y = 0 when x=2x = 2
  13. Ex 9.3 Q13
    Find a particular solution of the differential equation satisfying the given condition: cos⁡(dydx)=a (a∈R)\cos\left(\frac{dy}{dx}\right) = a\ (a \in \mathbf{R}); y=1y = 1 when x=0x = 0
  14. Ex 9.3 Q14
    Find a particular solution of the differential equation satisfying the given condition: dydx=ytan⁡x\frac{dy}{dx} = y \tan x; y=1y = 1 when x=0x = 0
  15. Ex 9.3 Q15
    Find the equation of a curve passing through the point (0,0)(0, 0) and whose differential equation is y′=exsin⁡xy' = e^x \sin x.
  16. Ex 9.3 Q16
    For the differential equation xydydx=(x+2)(y+2)xy\frac{dy}{dx} = (x + 2)(y + 2), find the solution curve passing through the point (1,−1)(1, -1).
  17. Ex 9.3 Q17
    Find the equation of a curve passing through the point (0,−2)(0, -2) given that at any point (x,y)(x, y) on the curve, the product of the slope of its tangent and yy coordinate of the point is equal to the xx coordinate of the point.
  18. Ex 9.3 Q18
    At any point (x,y)(x, y) of a curve, the slope of the tangent is twice the slope of the line segment joining the point of contact to the point (−4,−3)(-4, -3). Find the equation of the curve given that it passes through (−2,1)(-2, 1).
  19. Ex 9.3 Q19
    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 balloon after tt seconds.
  20. Ex 9.3 Q20
    In a bank, principal increases continuously at the rate of r%r\% per year. Find the value of rr if Rs 100 double itself in 10 years (log⁡e2=0.6931)(\log_e 2 = 0.6931).
  21. Ex 9.3 Q21
    In a bank, principal increases continuously at the rate of 5% per year. An amount of Rs 1000 is deposited with this bank, how much will it worth after 10 years (e0.5=1.648)(e^{0.5} = 1.648).
  22. Ex 9.3 Q22
    In a culture, the bacteria count is 1,00,000. The number is increased by 10% in 2 hours. In how many hours will the count reach 2,00,000, if the rate of growth of bacteria is proportional to the number present?
  23. Ex 9.3 Q23
    The general solution of the differential equation dydx=ex+y\frac{dy}{dx} = e^{x+y} is
    1. A.
      ex+e−y=Ce^x + e^{-y} = \mathrm{C}
    2. B.
      ex+ey=Ce^x + e^{y} = \mathrm{C}
    3. C.
      e−x+ey=Ce^{-x} + e^{y} = \mathrm{C}
    4. D.
      e−x+e−y=Ce^{-x} + e^{-y} = \mathrm{C}

9.4.2 Homogeneous Differential Equations

21 q

Solved Examples

Worked · 4
  1. 9.4 Eg.10
    Show that the differential equation (x−y)dydx=x+2y(x - y)\frac{dy}{dx} = x + 2y is homogeneous and solve it.
  2. 9.4 Eg.11
    Show that the differential equation xcos⁡(yx)dydx=ycos⁡(yx)+xx\cos\left(\frac{y}{x}\right)\frac{dy}{dx} = y\cos\left(\frac{y}{x}\right) + x is homogeneous and solve it.
  3. 9.4 Eg.12
    Show that the differential equation 2y exydx+(y−2x exy)dy=02y\,e^{\frac{x}{y}}dx + \left(y - 2x\,e^{\frac{x}{y}}\right)dy = 0 is homogeneous and find its particular solution, given that x=0x = 0 when y=1y = 1.
  4. 9.4 Eg.13
    Show that the family of curves for which the slope of the tangent at any point (x,y)(x, y) on it is x2+y22xy\frac{x^{2} + y^{2}}{2xy}, is given by x2−y2=cxx^{2} - y^{2} = cx.

Exercise 9.4

Practice · 17
  1. Ex 9.4 Q1
    Show that the given differential equation is homogeneous and solve it: (x2+xy) dy=(x2+y2) dx(x^{2} + xy)\,dy = (x^{2} + y^{2})\,dx.
  2. Ex 9.4 Q2
    Show that the given differential equation is homogeneous and solve it: y′=x+yxy' = \frac{x + y}{x}.
  3. Ex 9.4 Q3
    Show that the given differential equation is homogeneous and solve it: (x−y) dy−(x+y) dx=0(x - y)\,dy - (x + y)\,dx = 0.
  4. Ex 9.4 Q4
    Show that the given differential equation is homogeneous and solve it: (x2−y2) dx+2xy dy=0(x^{2} - y^{2})\,dx + 2xy\,dy = 0.
  5. Ex 9.4 Q5
    Show that the given differential equation is homogeneous and solve it: x2dydx=x2−2y2+xyx^{2}\frac{dy}{dx} = x^{2} - 2y^{2} + xy.
  6. Ex 9.4 Q6
    Show that the given differential equation is homogeneous and solve it: x dy−y dx=x2+y2 dxx\,dy - y\,dx = \sqrt{x^{2} + y^{2}}\,dx.
  7. Ex 9.4 Q7
    Show that the given differential equation is homogeneous and solve it: {xcos⁡(yx)+ysin⁡(yx)}y dx={ysin⁡(yx)−xcos⁡(yx)}x dy\left\{x\cos\left(\frac{y}{x}\right) + y\sin\left(\frac{y}{x}\right)\right\}y\,dx = \left\{y\sin\left(\frac{y}{x}\right) - x\cos\left(\frac{y}{x}\right)\right\}x\,dy.
  8. Ex 9.4 Q8
    Show that the given differential equation is homogeneous and solve it: xdydx−y+xsin⁡(yx)=0x\frac{dy}{dx} - y + x\sin\left(\frac{y}{x}\right) = 0.
  9. Ex 9.4 Q9
    Show that the given differential equation is homogeneous and solve it: y dx+xlog⁡(yx)dy−2x dy=0y\,dx + x\log\left(\frac{y}{x}\right)dy - 2x\,dy = 0.
  10. Ex 9.4 Q10
    Show that the given differential equation is homogeneous and solve it: (1+exy)dx+exy(1−xy)dy=0\left(1 + e^{\frac{x}{y}}\right)dx + e^{\frac{x}{y}}\left(1 - \frac{x}{y}\right)dy = 0.
  11. Ex 9.4 Q11
    Find the particular solution of the differential equation satisfying the given condition: (x+y) dy+(x−y) dx=0(x + y)\,dy + (x - y)\,dx = 0; y=1y = 1 when x=1x = 1.
  12. Ex 9.4 Q12
    Find the particular solution of the differential equation satisfying the given condition: x2 dy+(xy+y2) dx=0x^{2}\,dy + (xy + y^{2})\,dx = 0; y=1y = 1 when x=1x = 1.
  13. Ex 9.4 Q13
    Find the particular solution of the differential equation satisfying the given condition: [xsin⁡2(yx)−y]dx+x dy=0\left[x\sin^{2}\left(\frac{y}{x}\right) - y\right]dx + x\,dy = 0; y=π4y = \frac{\pi}{4} when x=1x = 1.
  14. Ex 9.4 Q14
    Find the particular solution of the differential equation satisfying the given condition: dydx−yx+cosec(yx)=0\frac{dy}{dx} - \frac{y}{x} + \text{cosec}\left(\frac{y}{x}\right) = 0; y=0y = 0 when x=1x = 1.
  15. Ex 9.4 Q15
    Find the particular solution of the differential equation satisfying the given condition: 2xy+y2−2x2dydx=02xy + y^{2} - 2x^{2}\frac{dy}{dx} = 0; y=2y = 2 when x=1x = 1.
  16. Ex 9.4 Q16
    A homogeneous differential equation of the from dxdy=h(xy)\frac{dx}{dy} = h\left(\frac{x}{y}\right) can be solved by making the substitution.
    1. A.
      y=vxy = vx
    2. B.
      v=yxv = yx
    3. C.
      x=vyx = vy
    4. D.
      x=vx = v
  17. Ex 9.4 Q17
    Which of the following is a homogeneous differential equation?
    1. A.
      (4x+6y+5) dy−(3y+2x+4) dx=0(4x + 6y + 5)\,dy - (3y + 2x + 4)\,dx = 0
    2. B.
      (xy) dx−(x3+y3) dy=0(xy)\,dx - (x^{3} + y^{3})\,dy = 0
    3. C.
      (x3+2y2) dx+2xy dy=0(x^{3} + 2y^{2})\,dx + 2xy\,dy = 0
    4. D.
      y2 dx+(x2−xy−y2) dy=0y^{2}\,dx + (x^{2} - xy - y^{2})\,dy = 0

9.4.3 Linear Differential Equations

24 q

Solved Examples

Worked · 5
  1. 9.5 Eg.14
    Find the general solution of the differential equation dydx−y=cos⁡x\frac{dy}{dx} - y = \cos x.
  2. 9.5 Eg.15
    Find the general solution of the differential equation xdydx+2y=x2x\frac{dy}{dx} + 2y = x^2 (x≠0)(x \neq 0).
  3. 9.5 Eg.16
    Find the general solution of the differential equation y dx−(x+2y2) dy=0y\,dx - (x + 2y^2)\,dy = 0.
  4. 9.5 Eg.17
    Find the particular solution of the differential equation dydx+ycot⁡x=2x+x2cot⁡x\frac{dy}{dx} + y\cot x = 2x + x^2\cot x (x≠0)(x \neq 0), given that y=0y = 0 when x=π2x = \frac{\pi}{2}.
  5. 9.5 Eg.18
    Find the equation of a curve passing through the point (0,1)(0, 1). If the slope of the tangent to the curve at any point (x,y)(x, y) is equal to the sum of the xx coordinate (abscissa) and the product of the xx coordinate and yy coordinate (ordinate) of that point.

Exercise 9.5

Practice · 19
  1. Ex 9.5 Q1
    Find the general solution of the differential equation dydx+2y=sin⁡x\frac{dy}{dx} + 2y = \sin x.
  2. Ex 9.5 Q2
    Find the general solution of the differential equation dydx+3y=e−2x\frac{dy}{dx} + 3y = e^{-2x}.
  3. Ex 9.5 Q3
    Find the general solution of the differential equation dydx+yx=x2\frac{dy}{dx} + \frac{y}{x} = x^2.
  4. Ex 9.5 Q4
    Find the general solution of the differential equation dydx+(sec⁡x) y=tan⁡x\frac{dy}{dx} + (\sec x)\,y = \tan x (0≤x<π2)\left(0 \le x < \frac{\pi}{2}\right).
  5. Ex 9.5 Q5
    Find the general solution of the differential equation cos⁡2x dydx+y=tan⁡x\cos^2 x\,\frac{dy}{dx} + y = \tan x (0≤x<π2)\left(0 \le x < \frac{\pi}{2}\right).
  6. Ex 9.5 Q6
    Find the general solution of the differential equation xdydx+2y=x2log⁡xx\frac{dy}{dx} + 2y = x^2\log x.
  7. Ex 9.5 Q7
    Find the general solution of the differential equation xlog⁡x dydx+y=2xlog⁡xx\log x\,\frac{dy}{dx} + y = \frac{2}{x}\log x.
  8. Ex 9.5 Q8
    Find the general solution of the differential equation (1+x2) dy+2xy dx=cot⁡x dx(1 + x^2)\,dy + 2xy\,dx = \cot x\,dx (x≠0)(x \neq 0).
  9. Ex 9.5 Q9
    Find the general solution of the differential equation xdydx+y−x+xycot⁡x=0x\frac{dy}{dx} + y - x + xy\cot x = 0 (x≠0)(x \neq 0).
  10. Ex 9.5 Q10
    Find the general solution of the differential equation (x+y)dydx=1(x + y)\frac{dy}{dx} = 1.
  11. Ex 9.5 Q11
    Find the general solution of the differential equation y dx+(x−y2) dy=0y\,dx + (x - y^2)\,dy = 0.
  12. Ex 9.5 Q12
    Find the general solution of the differential equation (x+3y2)dydx=y(x + 3y^2)\frac{dy}{dx} = y (y>0)(y > 0).
  13. Ex 9.5 Q13
    Find a particular solution of the differential equation dydx+2ytan⁡x=sin⁡x\frac{dy}{dx} + 2y\tan x = \sin x satisfying the given condition: y=0y = 0 when x=π3x = \frac{\pi}{3}.
  14. Ex 9.5 Q14
    Find a particular solution of the differential equation (1+x2)dydx+2xy=11+x2(1 + x^2)\frac{dy}{dx} + 2xy = \frac{1}{1 + x^2} satisfying the given condition: y=0y = 0 when x=1x = 1.
  15. Ex 9.5 Q15
    Find a particular solution of the differential equation dydx−3ycot⁡x=sin⁡2x\frac{dy}{dx} - 3y\cot x = \sin 2x satisfying the given condition: y=2y = 2 when x=π2x = \frac{\pi}{2}.
  16. Ex 9.5 Q16
    Find the equation of a curve passing through the origin given that the slope of the tangent to the curve at any point (x,y)(x, y) is equal to the sum of the coordinates of the point.
  17. Ex 9.5 Q17
    Find the equation of a curve passing through the point (0,2)(0, 2) given that the sum of the coordinates of any point on the curve exceeds the magnitude of the slope of the tangent to the curve at that point by 5.
  18. Ex 9.5 Q18
    The Integrating Factor of the differential equation xdydx−y=2x2x\frac{dy}{dx} - y = 2x^2 is
    1. A.
      e−xe^{-x}
    2. B.
      e−ye^{-y}
    3. C.
      1x\frac{1}{x}
    4. D.
      xx
  19. Ex 9.5 Q19
    The Integrating Factor of the differential equation (1−y2)dxdy+yx=ay(1 - y^2)\frac{dx}{dy} + yx = ay (−1<y<1)(-1 < y < 1) is
    1. A.
      1y2−1\frac{1}{y^2 - 1}
    2. B.
      1y2−1\frac{1}{\sqrt{y^2 - 1}}
    3. C.
      11−y2\frac{1}{1 - y^2}
    4. D.
      11−y2\frac{1}{\sqrt{1 - y^2}}

Miscellaneous Exercise on Chapter 9

24 q

Solved Examples

Worked · 4
  1. Misc Eg.19
    Verify that the function y=c1eaxcos⁡bx+c2eaxsin⁡bxy = c_1 e^{ax} \cos bx + c_2 e^{ax} \sin bx, where c1,c2c_1, c_2 are arbitrary constants is a solution of the differential equation d2ydx2−2adydx+(a2+b2)y=0\frac{d^2y}{dx^2} - 2a\frac{dy}{dx} + (a^2 + b^2)y = 0.
  2. Misc Eg.20
    Find the particular solution of the differential equation log⁡(dydx)=3x+4y\log\left(\frac{dy}{dx}\right) = 3x + 4y given that y=0y = 0 when x=0x = 0.
  3. Misc Eg.21
    Solve the differential equation (x dy−y dx) ysin⁡(yx)=(y dx+x dy) xcos⁡(yx)(x\,dy - y\,dx)\,y\sin\left(\frac{y}{x}\right) = (y\,dx + x\,dy)\,x\cos\left(\frac{y}{x}\right).
  4. Misc Eg.22
    Solve the differential equation (tan⁡−1y−x) dy=(1+y2) dx(\tan^{-1}y - x)\,dy = (1 + y^2)\,dx.

Miscellaneous Exercise

Practice · 20
  1. For each of the differential equations given below, indicate its order and degree (if defined).
    Misc Q1(i)
    d2ydx2+5x(dydx)2−6y=log⁡x\frac{d^2y}{dx^2} + 5x\left(\frac{dy}{dx}\right)^2 - 6y = \log x
  2. Misc Q1(ii)
    (dydx)3−4(dydx)2+7y=sin⁡x\left(\frac{dy}{dx}\right)^3 - 4\left(\frac{dy}{dx}\right)^2 + 7y = \sin x
  3. Misc Q1(iii)
    d4ydx4−sin⁡(d3ydx3)=0\frac{d^4y}{dx^4} - \sin\left(\frac{d^3y}{dx^3}\right) = 0
  4. For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.
    Misc Q2(i)
    xy=a ex+b e−x+x2xy = a\,e^{x} + b\,e^{-x} + x^2 : xd2ydx2+2dydx−xy+x2−2=0x\frac{d^2y}{dx^2} + 2\frac{dy}{dx} - xy + x^2 - 2 = 0
  5. Misc Q2(ii)
    y=ex(acos⁡x+bsin⁡x)y = e^{x}(a\cos x + b\sin x) : d2ydx2−2dydx+2y=0\frac{d^2y}{dx^2} - 2\frac{dy}{dx} + 2y = 0
  6. Misc Q2(iii)
    y=xsin⁡3xy = x\sin 3x : d2ydx2+9y−6cos⁡3x=0\frac{d^2y}{dx^2} + 9y - 6\cos 3x = 0
  7. Misc Q2(iv)
    x2=2y2log⁡yx^2 = 2y^2\log y : (x2+y2)dydx−xy=0(x^2 + y^2)\frac{dy}{dx} - xy = 0
  8. Misc Q3
    Prove that x2−y2=c (x2+y2)2x^2 - y^2 = c\,(x^2 + y^2)^2 is the general solution of differential equation (x3−3x y2) dx=(y3−3x2y) dy(x^3 - 3x\,y^2)\,dx = (y^3 - 3x^2y)\,dy, where cc is a parameter.
  9. Misc Q4
    Find the general solution of the differential equation dydx+1−y21−x2=0\frac{dy}{dx} + \sqrt{\frac{1 - y^2}{1 - x^2}} = 0.
  10. Misc Q5
    Show that the general solution of the differential equation dydx+y2+y+1x2+x+1=0\frac{dy}{dx} + \frac{y^2 + y + 1}{x^2 + x + 1} = 0 is given by (x+y+1)=A (1−x−y−2xy)(x + y + 1) = \mathrm{A}\,(1 - x - y - 2xy), where A\mathrm{A} is parameter.
  11. Misc Q6
    Find the equation of the curve passing through the point (0,π4)\left(0, \frac{\pi}{4}\right) whose differential equation is sin⁡xcos⁡y dx+cos⁡xsin⁡y dy=0\sin x \cos y\,dx + \cos x \sin y\,dy = 0.
  12. Misc Q7
    Find the particular solution of the differential equation (1+e2x) dy+(1+y2) ex dx=0(1 + e^{2x})\,dy + (1 + y^2)\,e^{x}\,dx = 0, given that y=1y = 1 when x=0x = 0.
  13. Misc Q8
    Solve the differential equation y exy dx=(x exy+y2)dy  (y≠0)y\,e^{\frac{x}{y}}\,dx = \left(x\,e^{\frac{x}{y}} + y^2\right)dy\;(y \ne 0).
  14. Misc Q9
    Find a particular solution of the differential equation (x−y) (dx+dy)=dx−dy(x - y)\,(dx + dy) = dx - dy, given that y=−1y = -1, when x=0x = 0. (Hint: put x−y=tx - y = t)
  15. Misc Q10
    Solve the differential equation [e−2xx−yx]dxdy=1  (x≠0)\left[\frac{e^{-2\sqrt{x}}}{\sqrt{x}} - \frac{y}{\sqrt{x}}\right]\frac{dx}{dy} = 1\;(x \ne 0).
  16. Misc Q11
    Find a particular solution of the differential equation dydx+ycot⁡x=4xcosec⁡x  (x≠0)\frac{dy}{dx} + y\cot x = 4x\operatorname{cosec} x\;(x \ne 0), given that y=0y = 0 when x=π2x = \frac{\pi}{2}.
  17. Misc Q12
    Find a particular solution of the differential equation (x+1)dydx=2e−y−1(x + 1)\frac{dy}{dx} = 2e^{-y} - 1, given that y=0y = 0 when x=0x = 0.
  18. Misc Q13
    The general solution of the differential equation y dx−x dyy=0\frac{y\,dx - x\,dy}{y} = 0 is
    1. A.
      xy=Cxy = \mathrm{C}
    2. B.
      x=Cy2x = \mathrm{C}y^2
    3. C.
      y=Cxy = \mathrm{C}x
    4. D.
      y=Cx2y = \mathrm{C}x^2
  19. Misc Q14
    The general solution of a differential equation of the type dxdy+P1x=Q1\frac{dx}{dy} + \mathrm{P}_1 x = \mathrm{Q}_1 is
    1. A.
      y e∫P1 dy=∫(Q1e∫P1 dy)dy+Cy\,e^{\int \mathrm{P}_1\,dy} = \int\left(\mathrm{Q}_1 e^{\int \mathrm{P}_1\,dy}\right)dy + \mathrm{C}
    2. B.
      y⋅e∫P1 dx=∫(Q1e∫P1 dx)dx+Cy \cdot e^{\int \mathrm{P}_1\,dx} = \int\left(\mathrm{Q}_1 e^{\int \mathrm{P}_1\,dx}\right)dx + \mathrm{C}
    3. C.
      x e∫P1 dy=∫(Q1e∫P1 dy)dy+Cx\,e^{\int \mathrm{P}_1\,dy} = \int\left(\mathrm{Q}_1 e^{\int \mathrm{P}_1\,dy}\right)dy + \mathrm{C}
    4. D.
      x e∫P1 dx=∫(Q1e∫P1 dx)dx+Cx\,e^{\int \mathrm{P}_1\,dx} = \int\left(\mathrm{Q}_1 e^{\int \mathrm{P}_1\,dx}\right)dx + \mathrm{C}
  20. Misc Q15
    The general solution of the differential equation ex dy+(y ex+2x) dx=0e^{x}\,dy + (y\,e^{x} + 2x)\,dx = 0 is
    1. A.
      x ey+x2=Cx\,e^{y} + x^2 = \mathrm{C}
    2. B.
      x ey+y2=Cx\,e^{y} + y^2 = \mathrm{C}
    3. C.
      y ex+x2=Cy\,e^{x} + x^2 = \mathrm{C}
    4. D.
      y ey+x2=Cy\,e^{y} + x^2 = \mathrm{C}