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Bernoulli equation: divide by a power of y and substitute

This type of question, asked 5 times: 2021 Paper 4, 2021 Paper 10, 2021 Paper 26, 25 Jan 2023, 12 Apr 2023. All 5 questions below, oldest first.

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  1. Question 1

    2021 Paper 4

    90
    If the curve y=y(x)y=y(x) is the solution of the differential equation (2xy2−y) dx+x dy=0(2xy^2-y)\,dx + x\,dy = 0 which passes through the point of intersection of the lines 2x−3y=12x-3y=1 and 3x+2y=83x+2y=8, then ∣y(1)∣|y(1)| is equal to

  2. Question 2

    2021 Paper 10

    61
    Let y=y(x)y = y(x) be the solution of the differential equation dydx=(y+1)((y+1)ex2/2−x),0<x<2.1\frac{dy}{dx}= (y + 1)\left( (y + 1)e^{x^{2}/2}- x \right),0 < x < 2.1, with y(2)=0y(2) = 0. Then the value of dydx\frac{dy}{dx} at x=1x = 1 is equal to:

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  3. Question 3

    2021 Paper 26

    66
    Let slope of the tangent line to a curve at any point P(x,y)P(x,y) be given by xy2+yx\frac{xy^{2}+y}{x}. If the curve intersects the line x+2y=4x+ 2y= 4 at x=−2x= - 2, then the value of yy, for which the point (3,y)(3,y) lies on the curve, is:

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  4. Question 4

    25 Jan 2023

    80
    Let y=y(x)y = y(x) be the solution curve of the differential equation dydx=yx(1+xy2(1+log⁡ex)),x>0,y(1)=3\frac{dy}{dx}=\frac{y}{x}\left( 1 + xy^{2}\left( 1 +\log_{e}x \right) \right),x > 0,y(1) = 3. Then y2(x)9\frac{y^{2}(x)}{9} is equal to :

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  5. Question 5

    12 Apr 2023

    69
    Let y=y(x),y>0y = y(x),y > 0, be a solution curve of the differential equation (1+x2)dy=y(x−y)dx\left( 1 +x^{2} \right)dy = y(x - y)dx. If y(0)=1y(0) = 1 and y(22)=βy(2\sqrt{2}) = \beta, then

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