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Linear equation with x as a function of y: integrating factor in y

This type of question, asked 6 times: 2021 Paper 21, 14 Jun 2022, 27 Jun 2022, 8 Apr 2023, 15 Apr 2023, 22 Jan 2025. All 7 questions below, oldest first.

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  1. Question 1

    2021 Paper 21

    68
    If the solution curve of the differential equation (2x−10y3)dy+ydx=0\left( 2x- 10y^{3} \right)dy+ydx= 0, passes through the points (0,1)(0,1) and (2,β)(2,\beta), then β\beta is a root of the equation:

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  2. Question 2

    14 Jun 2022

    74
    If x=x(y)x = x(y) is the solution of the differential equation ydxdy=2x+y3(y+1)ey,x(1)=0y\frac{dx}{dy}= 2x +y^{3}(y + 1)e^{y},x(1) = 0; then x(e)x(e) is equal to :

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  3. Question 3

    27 Jun 2022

    160
    If the solution curve of the differential equation ((tan⁡−1y)−x)dy=(1+y2)dx\left( \left( \tan^{- 1}y \right)- x \right)dy =\left( 1 +y^{2} \right)dx passes through the point (1,0)(1,0) then the abscissa of the point on the curve whose ordinate is tan⁡(1)\tan(1) is:

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  4. Question 4

    8 Apr 2023

    177
    Let the solution curve x=x(y),0<y<π2x=x(y),0 <y<\frac{\pi}{2}, of the differential equation (log⁡e(cos⁡y))2cos⁡ydx−(1+3x\left( \log_{e}(\cos y) \right)^{2}\cos ydx - (1 + 3x  log⁡e(cos⁡y))sin⁡ydy=0\left. \ \log_{e}(\cos y) \right)\sin ydy = 0 satisfy x(π3)=12log⁡e2x\left( \frac{\pi}{3} \right)=\frac{1}{2\log_{e}2}. If x(π6)=1log⁡em−log⁡enx\left( \frac{\pi}{6} \right)=\frac{1}{\log_{e}m-\log_{e}n}, where mm and nn are co-prime, then mnmn is equal to

  5. Question 5

    15 Apr 2023

    65
    Let x=x(y)x = x(y) be the solution of the differential equation 2(y+2)log⁡e(y+2)dx+(x+4−2log⁡e(y+2))dy=02(y + 2)\log_{e}(y + 2)dx +\left( x + 4 - 2\log_{e}(y + 2) \right)dy = 0, y>−1y > - 1 with x(e4−2)=1x\left( e^{4}- 2 \right)= 1. Then x(e9−2)x\left( e^{9}- 2 \right) is equal to

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  6. Question 6

    22 Jan 2025

    135
    If x=f(y)x = f(y) is the solution of the differential equation
    (1+y2)+(x−2etan⁡−1y)dydx=0,y∈(−π2,π2)\left( 1 +y^{2} \right)+\left( x - 2e^{\tan^{- 1}y} \right)\frac{dy}{dx}= 0,y \in\left( -\frac{\pi}{2},\frac{\pi}{2} \right)
    with f(0)=1f(0) = 1, then f(13)f\left( \frac{1}{\sqrt{3}} \right) is equal to :

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  7. Question 7

    22 Jan 2025

    62
    Let x=x(y)x = x(y) be the solution of the differential equation y2dx+(x−1y)dy=0y^{2}dx +\left( x -\frac{1}{y} \right)dy = 0. If x(1)=1x(1) = 1, then x(12)x\left( \frac{1}{2} \right) is :

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