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Linear equation dy/dx + P y = Q: integrating factor, then y at a point

This type of question, asked 41 times: 2021 Paper 3, 2021 Paper 6, 2021 Paper 8, 2021 Paper 12, 2021 Paper 17, 2021 Paper 20, 2021 Paper 24, 26 Jun 2022, 28 Jun 2022, 29 Jun 2022, 27 Jul 2022, 28 Jul 2022, 29 Jul 2022, 24 Jan 2023, 29 Jan 2023, 30 Jan 2023, 1 Feb 2023, 8 Apr 2023, 11 Apr 2023, 13 Apr 2023, 29 Jan 2024, 30 Jan 2024, 31 Jan 2024, 1 Feb 2024, 4 Apr 2024, 5 Apr 2024, 6 Apr 2024, 24 Jan 2025, 28 Jan 2025, 29 Jan 2025, 2 Apr 2025, 3 Apr 2025, 7 Apr 2025, 21 Jan 2026 S1, 21 Jan 2026 S2, 23 Jan 2026 S1, 28 Jan 2026 S2, 4 Apr 2026 S2, 5 Apr 2026 S2, 6 Apr 2026 S1, 8 Apr 2026 S2. All 49 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    2021 Paper 3

    78
    If a curve passes through the origin and the slope of the tangent to it at any point (x,y)(x,y) is x2−4x+y+8x−2\frac{x^{2}- 4x + y + 8}{x - 2}, then this curve also passes through the point:

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  2. Question 2

    2021 Paper 6

    67
    If y=y(x)y=y(x) is the solution of the differential equation dydx+(tan⁡x)y=sin⁡x,0≤x≤π3\frac{dy}{dx}+ (\tan x)y= \sin x,0 \leq x\leq\frac{\pi}{3}, with y(0)=0y(0) = 0, then y(π4)y\left( \frac{\pi}{4} \right) equal to :

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  3. Question 3

    2021 Paper 8

    63
    Let y=y(x)y=y(x) be the solution of the differential equation cos⁡x(3sin⁡x+cos⁡x+3)dy=\cos x(3\sin x + \cos x + 3)dy = (1+ysinx(3sin⁡x+cos⁡x+3))dx(1 +ysinx(3\sin x+ \cos x+ 3))dx, 0≤x≤π2,y(0)=00 \leq x\leq\frac{\pi}{2},y(0) = 0. Then, y(π3)y\left( \frac{\pi}{3} \right) is equal to:

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  4. Question 4

    2021 Paper 8

    70
    If the curve y=y(x)y=y(x) is the solution of the differential equation 2(x2+x5/4)dy−y(x+x1/4)dx=2x9/4dx,x>02\left( x^{2}+x^{5/4} \right)dy-y\left( x+x^{1/4} \right)dx= 2x^{9/4}dx,x> 0 which passes through the point (1,1−43log⁡e2)\left( 1,1 -\frac{4}{3}\log_{e}2 \right), then the value of y(16)y(16) is equal to:

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  5. Question 5

    2021 Paper 12

    51
    Let y=y(x)y=y(x) satisfies the equation dydx−∣A∣=0\frac{dy}{dx}- |A| = 0, for all x>0x> 0, where A=[ysin⁡x10−11201x]A=\begin{bmatrix} y & \sin x & 1 \\ 0 & - 1 & 1 \\ 2 & 0 & \frac{1}{x} \end{bmatrix}. If y(π)=π+2y(\pi) =\pi+ 2, then the value of y(π2)y\left( \frac{\pi}{2} \right) is:

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  6. Question 6

    2021 Paper 17

    70
    Let y=y(x)y=y(x) be the solution of the differential equation (x−x3)dy=(y+yx2−3x4)dx, x>2\left(x-x^{3}\right)dy=\left(y+yx^{2}-3x^{4}\right)dx,\ x>2. If y(3)=3y(3)=3, then y(4)y(4) is equal to:

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  7. Question 7

    2021 Paper 20

    68
    Let y=y(x)y=y(x) be the solution of the differential equation dydx=2(y+2sin⁡x−5)x−2cos⁡x\frac{dy}{dx}= 2(y+ 2\sin x- 5)x- 2\cos x such that y(0)=7y(0) = 7. Then y(π)y(\pi) is equal to:

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  8. Question 8

    2021 Paper 24

    67
    If y=y(x)y = y(x) is the solution curve of the differential equation x2dy+(y−1x)dx=0 ;x>0x^{2}dy +\left( y -\frac{1}{x} \right)dx = 0\ ;x > 0 and y(1)=1y(1) = 1, then y(12)y\left( \frac{1}{2} \right) is equal to:

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  9. Question 9

    26 Jun 2022

    160
    If the solution of the differential equation
    dydx+ex(x2−2)y=(x2−2x)(x2−2)e2x  satisfies \frac{dy}{dx}+e^{x}\left( x^{2}- 2 \right)y =\left( x^{2}- 2x \right)\left( x^{2}- 2 \right)e^{2x}\ \text{~satisfies~}
    y(0)=0y(0) = 0, then the value of y(2)y(2) is

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  10. Question 10

    26 Jun 2022

    85
    Let the solution curve y=y(x)y=y(x) of the differential equation (4+x2)dy−2x(x2+3y+4)dx=0\left( 4 +x^{2} \right)dy- 2x\left( x^{2}+ 3y+ 4 \right)dx= 0 pass through the origin. Then y(2)y(2) is equal to

  11. Question 11

    28 Jun 2022

    71
    Let y=y(x)y = y(x) be the solution of the differential equation x(1−x2)dydx+(3x2y−y−4x3)=0,x>1x\left( 1 -x^{2} \right)\frac{dy}{dx}+\left( 3x^{2}y - y - 4x^{3} \right)= 0,x > 1, with y(2)=−2y(2) = - 2. Then y(3)y(3) is equal to

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  12. Question 12

    29 Jun 2022

    82
    Let y=y(x)y=y(x) be the solution of the differential equation
    dydx+2y2cos⁡4x−cos⁡2x=xetan⁡−1(2cot⁡2x),0<x<\frac{dy}{dx}+\frac{\sqrt{2}y}{2\cos^{4}x- \cos2x}=xe^{\tan^{- 1}(\sqrt{2}\cot2x)},0 <x<
    π/2\pi/2 with y(π4)=π232y\left( \frac{\pi}{4} \right)=\frac{\pi^{2}}{32}. If y(π3)=π218e−tan⁡−1(α)y\left( \frac{\pi}{3} \right)=\frac{\pi^{2}}{18}e^{-\tan^{- 1}(\alpha)}, then the value of 3α23\alpha^{2} is equal to

  13. Question 13

    29 Jun 2022

    172
    Let y=y(x),x>1y=y(x),x> 1, be the solution of the differential equation (x−1)dydx+2xy=1x−1(x- 1)\frac{dy}{dx}+ 2xy=\frac{1}{x- 1}, with y(2)=1+e42e4y(2) =\frac{1 +e^{4}}{2e^{4}}. If y(3)=eα+1βeαy(3) =\frac{e^{\alpha}+ 1}{\beta e^{\alpha}}. then the value of α+β\alpha+\beta is equal to

  14. Question 14

    27 Jul 2022

    85
    Let y=y(x)y=y(x) be the solution curve of the differential equation sin⁡(2x2)log⁡e(tan⁡x2)dy+(4xy−42xsin(x2−π4))dx=0\sin\left( 2x^{2} \right)\log_{e}\left( \tan x^{2} \right)dy+\left( 4xy- 4\sqrt{2}xsin\left( x^{2}-\frac{\pi}{4} \right) \right)dx= 0, 0<x<π20 <x<\sqrt{\frac{\pi}{2}}, which passes through the point (π6,1)\left( \sqrt{\frac{\pi}{6}},1 \right). Then ∣y(π3)∣\left| y\left( \sqrt{\frac{\pi}{3}} \right) \right| is equal to

  15. Question 15

    28 Jul 2022

    70
    If y=y(x),x∈(0,π2)y = y(x),x \in\left( 0,\frac{\pi}{2} \right) be the solution curve of the differential equation
    (sin⁡22x)dydx+(8sin⁡22x+2sin⁡4x)y= 2e−4x(2sin⁡2x+cos⁡2x),  with  y(π4)=e−π,\begin{matrix} & \left( \sin^{2}2x \right)\frac{dy}{dx}+\left( 8\sin^{2}2x + 2\sin4x \right)y = \\ & \ 2e^{- 4x}(2\sin2x + \cos2x),\ \text{~with~}\ y\left( \frac{\pi}{4} \right)=e^{- \pi}, \end{matrix}
    then y(π6)y\left( \frac{\pi}{6} \right) is equal to:

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  16. Question 16

    28 Jul 2022

    161
    Let y=y(x)y = y(x) be the solution curve of the differential equation dydx+1x2−1y=(x−1x+1)12\frac{dy}{dx}+\frac{1}{x^{2}- 1}y =\left( \frac{x - 1}{x + 1} \right)^{\frac{1}{2}}, x>1x > 1 passing through the point (2,13)\left( 2,\sqrt{\frac{1}{3}} \right). Then 7y(8)\sqrt{7}y(8) is equal to

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  17. Question 17

    29 Jul 2022

    160
    Let y=y(x)y = y(x) be the solution curve of the differential equation dydx+(2x2+11x+13x3+6x2+11x+6)\frac{dy}{dx}+\left( \frac{2x^{2}+ 11x + 13}{x^{3}+ 6x^{2}+ 11x + 6} \right) y=(x+3)x+1,x>−1y =\frac{(x + 3)}{x + 1},x > - 1, which passes through the point (0,1)(0,1). Then y(1)y(1) is equal to:

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  18. Question 18

    24 Jan 2023

    70
    Let y=y(x)y = y(x) be the solution of the differential equation x3dy+(xy−1)dx=0,x>0,y(12)=3−ex^{3}dy + (xy - 1)dx = 0,x > 0,y\left( \frac{1}{2} \right)= 3 - e. Then y(1)y(1) is equal to

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  19. Question 19

    29 Jan 2023

    165
    Let y=y(x)y = y(x) be the solution of the differential equation xlog⁡exdydx+y=x2log⁡ex,(x>1)x\log_{e}x\frac{dy}{dx}+ y =x^{2}\log_{e}x,(x > 1). If y(2)=2y(2) = 2, then y(e)y(e) is equal to

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  20. Question 20

    30 Jan 2023

    78
    Let the solution curve y=y(x)y = y(x) of the differential equation dydx−3x5tan⁡−1(x3)(1+x6)32y=2xexp{x3−tan⁡−1x3(1+x6)}\frac{dy}{dx}-\frac{3x^{5}\tan^{- 1}\left( x^{3} \right)}{\left( 1 +x^{6} \right)^{\frac{3}{2}}}y = 2xexp\left\{ \frac{x^{3}-\tan^{- 1}x^{3}}{\sqrt{\left( 1 +x^{6} \right)}} \right\} pass through the origin. Then y(1)y(1) is equal to:

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  21. Question 21

    1 Feb 2023

    61
    If y=y(x)y = y(x) is the solution curve of the differential equation dydx+ytanx=xsecx,0≤x≤π3,y(0)=1\frac{dy}{dx}+ ytanx = xsecx,0 \leq x \leq\frac{\pi}{3},y(0) = 1, then y(π6)y\left( \frac{\pi}{6} \right) is equal to

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  22. Question 22

    8 Apr 2023

    88
    If the solution curve of the differential equation (y−2log⁡ex)dx+(xlog⁡ex2)dy=0,x>1\left( y - 2\log_{e}x \right)dx +\left( x\log_{e}x^{2} \right)dy = 0,x > 1 passes through the points (e,43)\left( e,\frac{4}{3} \right) and (e4,α)\left( e^{4},\alpha \right), then α\alpha is equal to-

  23. Question 23

    11 Apr 2023

    155
    Let y=y(x)y = y(x) be the solution of the differential equations dydx+5x(x5+1)y=(x5+1)2x7,x>0\frac{dy}{dx}+\frac{5}{x\left( x^{5}+ 1 \right)}y =\frac{\left( x^{5}+ 1 \right)^{2}}{x^{7}},x > 0. If y(1)=2y(1) = 2, then y(2)y(2) is equal to

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  24. Question 24

    13 Apr 2023

    177
    If y=y(x)y = y(x) is the solution of the differential equation dydx+4x(x2−1)y=x+2(x2−1)52,x>1\frac{dy}{dx}+\frac{4x}{\left( x^{2}- 1 \right)}y =\frac{x + 2}{\left( x^{2}- 1 \right)^{\frac{5}{2}}},x > 1 such that y(2)=29log⁡e(2+3)y(2) =\frac{2}{9}\log_{e}(2 +\sqrt{3}) and  y(2)=\ y(\sqrt{2}) = αlog⁡e(α+β)+β−γ,α,β,γ∈N\alpha\log_{e}(\sqrt{\alpha}+ \beta) + \beta -\sqrt{\gamma},\alpha,\beta,\gamma \in N, then αβγ\alpha\beta\gamma is equal to

  25. Question 25

    29 Jan 2024

    72
    A function y=f(x)y = f(x) satisfies f(x)sin⁡2x+sin⁡x−(1+cos⁡2x)f′(x)=0f(x)\sin2x + \sin x -\left( 1 +\cos^{2}x \right)f^{'}(x) = 0 with condition f(0)=0f(0) = 0. Then f(π2)f\left( \frac{\pi}{2} \right) is equal to

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  26. Question 26

    30 Jan 2024

    87
    Let y=y(x)y = y(x) be the solution of the differential equation (1−x2)dy=[xy+(x3+2)3(1−x2)]dx\left( 1 -x^{2} \right)dy=\left\lbrack xy+\left( x^{3}+ 2 \right)\sqrt{3\left( 1 -x^{2} \right)} \right\rbrack dx, −1<x<1,y(0)=0- 1 <x< 1,y(0) = 0. If y(12)=mn,my\left( \frac{1}{2} \right)=\frac{m}{n},m and nn are coprime numbers, then m+nm + n is equal to

  27. Question 27

    31 Jan 2024

    71
    Let y=y(x)y = y(x) be the solution of the differential equation
    dydx=(tan⁡x)+ysin⁡x(secx−sin⁡xtanx)\frac{dy}{dx}=\frac{(\tan x) +y}{\sin x(secx- \sin xtanx)}
    x∈(0,π2)x \in\left( 0,\frac{\pi}{2} \right) satisfying the condition y(π4)=2y\left( \frac{\pi}{4} \right)= 2. Then, y(π3)y\left( \frac{\pi}{3} \right) is

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  28. Question 28

    1 Feb 2024

    81
    If x=x(t)x = x(t) is the solution of the differential equation (t+1)dx=(2x+(t+1)4)dt,x(0)=2(t+ 1)dx=\left( 2x+ (t+ 1)^{4} \right)dt,x(0) = 2, then, x(1)x(1) equals

  29. Question 29

    4 Apr 2024

    168
    Let y=y(x)y = y(x) be the solution of the differential equation (x2+4)2dy+(2x3y+8xy−2)dx=0\left( x^{2}+ 4 \right)^{2}dy +\left( 2x^{3}y + 8xy - 2 \right)dx = 0. If y(0)=0y(0) = 0 then y(2)y(2) is equal to

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  30. Question 30

    4 Apr 2024

    83
    Let the solution y=y(x)y = y(x) of the differential equation dydx−y=1+4sin⁡x\frac{dy}{dx}- y = 1 + 4\sin x satisfy y(π)=1y(\pi) = 1. Then y(π2)+10y\left( \frac{\pi}{2} \right)+ 10 is equal to

  31. Question 31

    5 Apr 2024

    64
    If y=y(x)y = y(x) is the solution of the differential equation dydx+2y=sin⁡(2x),y(0)=34\frac{dy}{dx}+ 2y = \sin(2x),y(0) =\frac{3}{4}, then y(π8)y\left( \frac{\pi}{8} \right) is equal to :

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  32. Question 32

    6 Apr 2024

    77
    Let y=y(x)y = y(x) be the solution of the differential equation (1+x2)dydx+y=etan⁡−1x,y(1)=0\left( 1 +x^{2} \right)\frac{dy}{dx}+ y =e^{\tan^{- 1}x},y(1) = 0. Then y(0)y(0) is

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  33. Question 33

    6 Apr 2024

    78
    Let y=y(x)y = y(x) be the solution of the differential equation (2xlog⁡ex)dydx+2y=3xlog⁡ex,x>0\left( 2x\log_{e}x \right)\frac{dy}{dx}+ 2y =\frac{3}{x}\log_{e}x,x > 0 and y(e−1)=0y\left( e^{- 1} \right)= 0. Then, y(e)y(e) is equal to

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  34. Question 34

    24 Jan 2025

    59
    Let y=y(x)y = y(x) be the solution of the differential equation (xy−5x21+x2)dx+(1+x2)dy=0\left( xy - 5x^{2}\sqrt{1 +x^{2}} \right)dx +\left( 1 +x^{2} \right)dy = 0, y(0)=0y(0) = 0. Then y(3)y(\sqrt{3}) is equal to

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  35. Question 35

    24 Jan 2025

    129
    Let f:(0,∞)→Rf:(0,\infty) \rightarrow R be a function which is differentiable at all points of its domain and satisfies the condition x2f′(x)=2xf(x)+3x^{2}f^{'}(x) = 2xf(x) + 3, with f(1)=4f(1) = 4. Then 2f(2)2f(2) is equal to:

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  36. Question 36

    28 Jan 2025

    150
    If y=y(x)y = y(x) is the solution of the differential equation, 4−x2dydx=((sin⁡−1(x2))2−y)sin⁡−1(x2)\sqrt{4 -x^{2}}\frac{dy}{dx}=\left( \left( \sin^{- 1}\left( \frac{x}{2} \right) \right)^{2}-y \right)\sin^{- 1}\left( \frac{x}{2} \right) −2≤x≤2,y(2)=(π2−84)- 2 \leq x\leq 2,y(2) =\left( \frac{\pi^{2}- 8}{4} \right), then y2(0)y^{2}(0) is equal to

  37. Question 37

    29 Jan 2025

    60
    Let y=y(x)y = y(x) be the solution of the differential equation cos⁡x(log⁡e(cos⁡x))2dy+(sin⁡x−3ysinxlog⁡e(cos⁡x))dx=0\cos x\left( \log_{e}(\cos x) \right)^{2}dy +\left( \sin x - 3ysinx\log_{e}(\cos x) \right)dx = 0, x∈(0,π2)x \in\left( 0,\frac{\pi}{2} \right). If y(π4)=−1log⁡e2y\left( \frac{\pi}{4} \right)=\frac{- 1}{\log_{e}2}, then y(π6)y\left( \frac{\pi}{6} \right) is :

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  38. Question 38

    29 Jan 2025

    145
    If for the solution curve y=f(x)y = f(x) of the differential equation dydx+(tan⁡x)y=2+secx(1+2secx)2\frac{dy}{dx}+ (\tan x)y =\frac{2 + secx}{(1 + 2secx)^{2}}, x∈(−π2,π2),f(π3)=310x \in\left( \frac{- \pi}{2},\frac{\pi}{2} \right),f\left( \frac{\pi}{3} \right)=\frac{\sqrt{3}}{10}, then f(π4)f\left( \frac{\pi}{4} \right) is equal to:

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  39. Question 39

    2 Apr 2025

    146
    Let y=y(x)y = y(x) be the solution of the differential equation dydx+2ysec⁡2x=2sec⁡2x+3tan⁡x⋅sec⁡2x\frac{dy}{dx} + 2y\sec^{2}x = 2\sec^{2}x + 3\tan x \cdot \sec^{2}x such that y(0)=54y(0) = \frac{5}{4}. Then 12(y(π4)−e−2)12\left( y\left( \frac{\pi}{4} \right) - e^{- 2} \right) is equal to

  40. Question 40

    3 Apr 2025

    139
    Let y=y(x)y=y\left( x \right) be the solution of the differential equation dydx+3(tan⁡2x)y+3y=\frac{dy}{dx}+ 3\left( \tan^{2}x \right)y+ 3y= sec⁡2x\sec^{2}x, y(0)=13+e3y(0) =\frac{1}{3}+e^{3}. Then y(π4)y\left( \frac{\pi}{4} \right) is equal to.

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  41. Question 41

    7 Apr 2025

    56
    Let y=y(x)y=y(x) be the solution curve of the differential equation
    x(x2+ex)dy+(ex(x−2)y−x3)dx=0,x>0x\left( x^{2}+e^{x} \right)dy+\left( e^{x}(x- 2)y-x^{3} \right)dx= 0,x> 0
    passing through the point (1,0)(1,0). Then y(2)y(2) is equal to :

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  42. Question 42

    21 Jan 2026 S1

    58
    Let y=y(x)y = y(x) be the solution curve of the differential equation (1+x2)dy+(y−tan⁡−1x)dx=0\left( 1 +x^{2} \right)dy +\left( y -\tan^{- 1}x \right)dx = 0, y(0)=1y(0) = 1. Then the value of y(1)y(1) is :

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  43. Question 43

    21 Jan 2026 S2

    66
    Let y=y(x)y = y(x) be the solution of the differential equation secxdydx−2y=2+3sin⁡x,x∈(−π2,π2)secx\frac{dy}{dx}- 2y = 2 + 3\sin x,x \in\left( -\frac{\pi}{2},\frac{\pi}{2} \right), y(0)=−74y(0) = -\frac{7}{4}. Then y(π6)y\left( \frac{\pi}{6} \right) is equal to:

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  44. Question 44

    23 Jan 2026 S1

    57
    Let y=y(x)y = y(x) be the solution of the differential equation x4dy+(4x3y+2sin⁡x)dx=0,x>0,y(π2)=0x^{4}dy +\left( 4x^{3}y + 2\sin x \right)dx = 0,x > 0,y\left( \frac{\pi}{2} \right)= 0. Then π4y(π3)\pi^{4}y\left( \frac{\pi}{3} \right) is equal to :

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  45. Question 45

    28 Jan 2026 S2

    64
    Let y=y(x)y=y(x) be the solution of the differential equation xdydx−y=x2cot⁡x,x∈(0,π)x\frac{dy}{dx}-y=x^{2}\cot x,x\in (0,\pi). If y(π2)=π2y\left( \frac{\pi}{2} \right)=\frac{\pi}{2}, then 6y(π6)−8y(π4)6y\left( \frac{\pi}{6} \right)- 8y\left( \frac{\pi}{4} \right) is equal to :

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  46. Question 46

    4 Apr 2026 S2

    69
    Let y=y(x)y=y(x) be the solution of the differential equation : dydx+(6x2+(3x2+2x3+4)e−2x(x3+2)(2+e−2x))y=2+e−2x\frac{dy}{dx}+\left( \frac{6x^{2}+\left( 3x^{2}+ 2x^{3}+ 4 \right)e^{- 2x}}{\left( x^{3}+ 2 \right)\left( 2 +e^{- 2x} \right)} \right)y= 2 +e^{- 2x},x∈(−1,2)x \in ( - 1,2), satisfying y(0)=32y(0) =\frac{3}{2}. If y(1)=α(2+e−2)y(1) =\alpha\left( 2 +e^{- 2} \right), then α\alpha is equal to :

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  47. Question 47

    5 Apr 2026 S2

    75
    Let y=y(x)y = y(x) be the solution of the differential equation (tan⁡x)1/2dy=(sec⁡3x−(tan⁡x)3/2y)dx(\tan x)^{1/2}dy = \left( \sec^{3}x - (\tan x)^{3/2}y \right)dx, 0<x<π2,y(π4)=6250 < x < \frac{\pi}{2},y\left( \frac{\pi}{4} \right) = \frac{6\sqrt{2}}{5}. If y(π3)=45αy\left( \frac{\pi}{3} \right) = \frac{4}{5}\alpha, then α4\alpha^{4} equals ____\_\_\_\_

  48. Question 48

    6 Apr 2026 S1

    75
    Let y=y(x)y = y(x) be the solution of the differential equation (x2−xx2−1)dy+(y(x−x2−1)−x)dx=0\left( x^{2} - x\sqrt{x^{2} - 1} \right)dy + \left( y\left( x - \sqrt{x^{2} - 1} \right) - x \right)dx = 0, x≥1x \geq 1. If y(1)=1y(1) = 1, then the greatest integer less than y(5)y(\sqrt{5}) is ____\_\_\_\_ .

  49. Question 49

    8 Apr 2026 S2

    67
    Let y=y(x)y=y(x) be the solution of the differential equation x1−x2dy+(y1−x2−xcos⁡−1x)dx=0x\sqrt{1 -x^{2}}dy+\left( y\sqrt{1 -x^{2}}-x\cos^{- 1}x \right)dx= 0, x∈(0,1),lim⁡x→1−y(x)=1x\in (0,1),\lim_{x\rightarrow1^{-}} y(x) = 1. Then y(12)y\left( \frac{1}{2} \right) equals :

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