PYQ Vault
EbooksTutor

Homogeneous equation: substitute y = vx and find the curve or a value

This type of question, asked 19 times: 2021 Paper 6, 2021 Paper 10, 2021 Paper 23, 25 Jun 2022, 28 Jun 2022, 29 Jun 2022, 25 Jul 2022, 28 Jul 2022, 29 Jul 2022, 24 Jan 2023, 30 Jan 2023, 31 Jan 2023, 10 Apr 2023, 27 Jan 2024, 31 Jan 2024, 9 Apr 2024, 23 Jan 2025, 22 Jan 2026 S1, 2 Apr 2026 S2. All 21 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    2021 Paper 6

    79
    Let C1C_{1} be the curve obtained by the solution of differential equation 2xydydx=y2−x2,x>02xy\frac{dy}{dx}=y^{2}-x^{2},x> 0. Let the curve C2C_{2} be the solution of 2xyx2−y2=dydx\frac{2xy}{x^{2}-y^{2}}=\frac{dy}{dx}. If both the curves pass through (1,1)(1,1), then the area enclosed by the curves C1C_{1} and C2C_{2} is equal to:

    Tap an option to check your answer.

  2. Question 2

    2021 Paper 10

    90
    Let y=y(x)y=y(x) be the solution of the differential equation x dy−y dx=x2−y2 dx, x≥1x\,dy-y\,dx=\sqrt{x^{2}-y^{2}}\,dx,\ x\geq 1, with y(1)=0y(1) = 0. If the area bounded by the lines x=1, x=eπ, y=0x = 1,\ x = e^{\pi},\ y = 0 and the curve y=y(x)y = y(x) is αe2π+β\alpha e^{2\pi}+\beta, then the value of 10(α+β)10(\alpha+\beta) is equal to

  3. Question 3

    2021 Paper 23

    72
    If ydydx=x[y2x2+ϕ(y2x2)ϕ′(y2x2)],x>0,ϕ>0y\frac{dy}{dx}=x\left\lbrack \frac{y^{2}}{x^{2}}+\frac{\phi\left( \frac{y^{2}}{x^{2}} \right)}{\phi^{'}\left( \frac{y^{2}}{x^{2}} \right)} \right\rbrack,x> 0,\phi> 0, and y(1)=−1y(1) = - 1, then ϕ(y24)\phi\left( \frac{y^{2}}{4} \right) is equal to:

    Tap an option to check your answer.

  4. Question 4

    25 Jun 2022

    163
    If y=y(x)y = y(x) is the solution of the differential equation 2x2dydx−2xy+3y2=02x^{2}\frac{dy}{dx}- 2xy + 3y^{2}= 0 such that y(e)=e3y(e) =\frac{e}{3}, then y(1)y(1) is equal to

    Tap an option to check your answer.

  5. Question 5

    25 Jun 2022

    78
    If the solution curve y=y(x)y = y(x) of the differential equation y2dx+(x2−xy+y2)dy=0y^{2}dx +\left( x^{2}- xy +y^{2} \right)dy = 0, which passes through the point (1,1)(1,1) and intersects the line y=3xy =\sqrt{3}x at the point (α,3α)(\alpha,\sqrt{3}\alpha), then value of log⁡e(3α)\log_{e}(\sqrt{3}\alpha) is equal to

    Tap an option to check your answer.

  6. Question 6

    28 Jun 2022

    70
    Let the solution curve y=y(x)y = y(x) of the differential equation,
    [xx2−y2+eyx]xdydx=x+[xx2−y2+eyx]y\left\lbrack \frac{x}{\sqrt{x^{2}-y^{2}}}+e^{\frac{y}{x}} \right\rbrack x\frac{dy}{dx}= x +\left\lbrack \frac{x}{\sqrt{x^{2}-y^{2}}}+e^{\frac{y}{x}} \right\rbrack y
    pass through the points (1,0)(1,0) and (2α,α),α>0(2\alpha,\alpha),\alpha > 0. Then α\alpha is equal to

    Tap an option to check your answer.

  7. Question 7

    29 Jun 2022

    62
    Let the solution curve of the differential equation xdydx−y=y2+16x2,y(1)=3x\frac{dy}{dx}- y =\sqrt{y^{2}+ 16x^{2}},y(1) = 3 be y=y(x)y = y(x). Then y(2)y(2) is equal to:

    Tap an option to check your answer.

  8. Question 8

    25 Jul 2022

    71
    The general solution of the differential equation (x−y2)dx+y(5x+y2)dy=0\left( x -y^{2} \right)dx + y\left( 5x +y^{2} \right)dy = 0 is :

    Tap an option to check your answer.

  9. Question 9

    25 Jul 2022

    175
    Let y=y(x)y = y(x) be the solution of the differential equation dydx=4y3+2yx23xy2+x3,y(1)=1\frac{dy}{dx}=\frac{4y^{3}+ 2yx^{2}}{3xy^{2}+x^{3}},y(1) = 1. If for some n∈N,y(2)∈[n−1,n)n \in N,y(2) \in \lbrack n - 1,n), then nn is equal to

  10. Question 10

    28 Jul 2022

    61
    Let the solution curve of the differential equation xdy=(x2+y2+y)dx,x>0,xdy =\left( \sqrt{x^{2}+y^{2}}+ y \right)dx,x > 0, intersect the line x=1x = 1 at y=0y = 0 and the line x=2x = 2 at y=αy = \alpha. Then the value of α\alpha is:

    Tap an option to check your answer.

  11. Question 11

    29 Jul 2022

    159
    If the solution curve of the differential equation dydx=x+y−2x−y\frac{dy}{dx}=\frac{x + y - 2}{x - y} passes through the point (2,1)(2,1) and (k+1,2),k>0(k + 1,2),k > 0, then

    Tap an option to check your answer.

  12. Question 12

    24 Jan 2023

    155
    Let y=y(x)y = y(x) be the solution of the differential equation (x2−3y2)dx+3xydy=0,y(1)=1\left( x^{2}- 3y^{2} \right)dx + 3xydy = 0,y(1) = 1. Then 6y2(e)6y^{2}(e) is equal to

    Tap an option to check your answer.

  13. Question 13

    30 Jan 2023

    167
    The solution of the differential equation dydx=−(x2+3y23x2+y2),y(1)=0\frac{dy}{dx}= -\left( \frac{x^{2}+ 3y^{2}}{3x^{2}+y^{2}} \right),y(1) = 0 is

    Tap an option to check your answer.

  14. Question 14

    31 Jan 2023

    155
    Let y=y(x)y = y(x) be the solution of the differential equation (3y2−5x2)y dx+2x(x2−y2)dy=0\left( 3y^{2}- 5x^{2} \right)y\text{ }dx + 2x\left( x^{2}-y^{2} \right)dy = 0 such that y(1)=1y(1) = 1. Then ∣(y(2))3−12y(2)∣\left| (y(2))^{3}- 12y(2) \right| is equal to:

    Tap an option to check your answer.

  15. Question 15

    10 Apr 2023

    70
    The slope of tangent at any point (x,y)(x,y) on a curve y=y(x)y = y(x) is x2+y22xy,x>0\frac{x^{2}+y^{2}}{2xy},x > 0. If y(2)=0y(2) = 0, then a value of y(8)y(8) is

    Tap an option to check your answer.

  16. Question 16

    27 Jan 2024

    176
    If the solution curve, of the differential equation dydx=x+y−2x−y\frac{dy}{dx}=\frac{x + y - 2}{x - y} passing through the point (2,1)(2,1) is tan⁡−1(y−1x−1)−1βlog⁡e(α+(y−1x−1)2)=log⁡e∣x−1∣\tan^{- 1}\left( \frac{y - 1}{x - 1} \right)-\frac{1}{\beta}\log_{e}\left( \alpha +\left( \frac{y - 1}{x - 1} \right)^{2} \right)=\log_{e}|x - 1|, then 5β+α5\beta + \alpha is equal to

  17. Question 17

    31 Jan 2024

    69
    The solution curve of the differential equation ydxdy=x(log⁡ex−log⁡ey+1),x>0,y>0y\frac{dx}{dy}=x\left( \log_{e}x-\log_{e}y+ 1 \right),x> 0,y> 0 passing through the point (e,1)(e,1) is

    Tap an option to check your answer.

  18. Question 18

    9 Apr 2024

    77
    The solution of the differential equation (x2+y2)dx−5xydy=0,y(1)=0\left( x^{2}+y^{2} \right)dx - 5xydy = 0,y(1) = 0, is :

    Tap an option to check your answer.

  19. Question 19

    23 Jan 2025

    136
    Let x=x(y)x = x(y) be the solution of the differential equation
    y=(x−ydxdy)sin⁡(xy),y>0 and x(1)=π2y =\left( x - y\frac{dx}{dy} \right)\sin\left( \frac{x}{y} \right),y > 0\text{~and~}x(1) =\frac{\pi}{2}
    Then cos⁡(x(2))\cos(x(2)) is equal to :

    Tap an option to check your answer.

  20. Question 20

    22 Jan 2026 S1

    69
    Let the solution curve of the differential equation xdy−ydx=x2+y2dx,x>0,y(1)=0xdy - ydx =\sqrt{x^{2}+y^{2}}dx,x > 0,y(1) = 0, be y=y(x)y = y(x). Then y(3)y(3) is equal to

    Tap an option to check your answer.

  21. Question 21

    2 Apr 2026 S2

    70
    Let x=x(y)x = x(y) be the solution of the differential equation 2y2dxdy−2xy+x2=0,y>1,x(e)=e2y^{2}\frac{dx}{dy}- 2xy+x^{2}= 0,y> 1,x(e) =e. Then x(e2)x\left( e^{2} \right) is equal to:

    Tap an option to check your answer.