PYQ Vault
EbooksTutor

Separate the variables and integrate: find y at a point

This type of question, asked 13 times: 2021 Paper 7, 2021 Paper 11, 2021 Paper 16, 2021 Paper 17, 29 Jan 2023, 27 Jan 2024, 30 Jan 2024, 4 Apr 2024, 6 Apr 2024, 9 Apr 2024, 23 Jan 2025, 22 Jan 2026 S2, 2 Apr 2026 S1. All 14 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    2021 Paper 7

    79
    Which of the following is true for y(x)y(x) that satisfies the differential equation? dydx=xy−1+x−y;y(0)=0\frac{dy}{dx}=xy- 1 +x-y;y(0) = 0 :

    Tap an option to check your answer.

  2. Question 2

    2021 Paper 11

    56
    Let y=y(x)y=y(x) be the solution of the differential equation ex1−y2dx+(yx)dy=0,y(1)=−1e^{x}\sqrt{1 -y^{2}}dx+\left( \frac{y}{x} \right)dy= 0,y(1) = - 1. Then the value of (y(3))2(y(3))^{2} is equal to:

    Tap an option to check your answer.

  3. Question 3

    2021 Paper 16

    61
    If secx(dydx)−sin⁡(x+y)−sin⁡(x−y)=0secx\left( \frac{dy}{dx} \right)- \sin(x+y) - \sin(x-y) = 0 and y(0)=0y(0) = 0. Find 5[y′(π2)]5\left\lbrack y^{'}\left( \frac{\pi}{2} \right) \right\rbrack.

  4. Question 4

    2021 Paper 17

    87
    Let y=y(x)y=y(x) be the solution of the differential equation dy=eαx+ydx;α∈Ndy =e^{\alpha x+y}dx;\alpha\in N. If y(log⁡e2)=log⁡e2y\left( \log_{e}2 \right)=\log_{e}2 and y(0)=log⁡e(12)y(0) =\log_{e}\left( \frac{1}{2} \right), then the value of α\alpha is equal to

  5. Question 5

    29 Jan 2023

    65
    Let y=f(x)y = f(x) be the solution of the differential equation y(x+1)dx−x2dy=0,y(1)=ey(x + 1)dx -x^{2}dy = 0,y(1) = e. Then lim⁡x→0+f(x)\lim_{x \rightarrow0^{+}} f(x) is equal to

    Tap an option to check your answer.

  6. Question 6

    27 Jan 2024

    167
    If y=y(x)y = y(x) is the solution curve of the differential equation (x2−4)dy−(y2−3y)dx=0\left( x^{2}- 4 \right)dy -\left( y^{2}- 3y \right)dx = 0, x>2,y(4)=32x > 2,y(4) =\frac{3}{2} and the slope of the curve is never zero, then the value of y(10)y(10) equals:

    Tap an option to check your answer.

  7. Question 7

    30 Jan 2024

    69
    Let y=y(x)y=y(x) be the solution of the differential equation sec⁡x dy+{2(1−x)tan⁡x+x(2−x)} dx=0\sec x\,dy + \{2(1-x)\tan x + x(2-x)\}\,dx = 0 such that y(0)=2y(0)=2. Then y(2)y(2) is equal to:

    Tap an option to check your answer.

  8. Question 8

    4 Apr 2024

    64
    If the solution y=y(x)y = y(x) of the differential equation (x4+2x3+3x2+2x+2)dy−(2x2+2x+3)dx=0\left( x^{4}+ 2x^{3}+ 3x^{2}+ 2x + 2 \right)dy -\left( 2x^{2}+ 2x + 3 \right)dx = 0 satisfies y(−1)=−π4y( - 1) = -\frac{\pi}{4}, then y(0)y(0) is equal to :

    Tap an option to check your answer.

  9. Question 9

    6 Apr 2024

    180
    If the solution y(x)y(x) of the given differential equation (ey+1)cos⁡xdx+eysin⁡xdy=0\left( e^{y}+ 1 \right)\cos xdx+e^{y}\sin xdy= 0 passes through the point (π2,0)\left( \frac{\pi}{2},0 \right), then the value of ey(π6)e^{y\left( \frac{\pi}{6} \right)} is equal to

  10. Question 10

    9 Apr 2024

    63
    The solution curve, of the differential equation 2ydydx+3=5dydx2y\frac{dy}{dx}+ 3 = 5\frac{dy}{dx}, passing through the point (0,1)(0,1) is a conic, whose vertex lies on the line :

    Tap an option to check your answer.

  11. Question 11

    23 Jan 2025

    55
    Let a curve y=f(x)y = f(x) pass through the points (0,5)(0,5) and (log⁡e2,k)\left( \log_{e}2,k \right). If the curve satisfies the differential equation 2(3+y)e2xdx−(7+e2x)dy=02(3 + y)e^{2x}dx -\left( 7 +e^{2x} \right)dy = 0, then kk is equal to

    Tap an option to check your answer.

  12. Question 12

    22 Jan 2026 S2

    56
    If y=y(x)y = y(x) satisfies the differential equation 16x 9+x (4+9+x)cos⁡ydy=(1+216\sqrt{x}\,\sqrt{9 + \sqrt{x}}\,(4 +\sqrt{9 +\sqrt{x}})\cos ydy = (1 + 2 siny)dx, x>0x > 0 and y(256)=π2,y(49)=αy(256) =\frac{\pi}{2},y(49) = \alpha, then 2 sin⁡α\sin\alpha is equal to :

    Tap an option to check your answer.

  13. Question 13

    2 Apr 2026 S1

    70
    Let y=y(x)y=y(x) be the solution curve of the differential equation (1+sin⁡x)dydx+(y+1)cos⁡x=0,y(0)=0(1 + \sin x)\frac{dy}{dx}+ (y+ 1)\cos x= 0,y(0) = 0. If the curve y=y(x)y = y(x) passes through the point (α,−12)\left( \alpha,\frac{- 1}{2} \right), then a value of α\alpha is :

    Tap an option to check your answer.

  14. Question 14

    2 Apr 2026 S1

    67
    If the curve y=f(x)y=f(x) passes through the point ( 1,e)1,e) and satisfies the differential equation dy=y(2+log⁡ex)dx,x>0dy=y\left( 2 +\log_{e}x \right)dx,x> 0, then f(e)f(e) is equal to :

    Tap an option to check your answer.