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Substitute v = ax + by + c to make the equation separable

This type of question, asked 3 times: 27 Jan 2024, 1 Feb 2024, 4 Apr 2024. All 3 questions below, oldest first.

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  1. Question 1

    27 Jan 2024

    83
    If the solution of the differential equation (2x+3y−2)dx+(4x+6y−7)dy=0,y(0)=3(2x+ 3y- 2)dx+ (4x+ 6y- 7)dy= 0,y(0) = 3, is αx+βy+3log⁡e∣2x+3y−γ∣=6\alpha x+\beta y+ 3\log_{e}|2x+ 3y-\gamma| = 6, then α+2β+3γ\alpha+ 2\beta+ 3\gamma is equal to

  2. Question 2

    1 Feb 2024

    74
    Let y=y(x)y = y(x) be the solution of the differential equation dydx=2x(x+y)3−x(x+y)−1,y(0)=1\frac{dy}{dx}= 2x(x + y)^{3}- x(x + y) - 1,y(0) = 1. Then, (12+y(12))2\left( \frac{1}{\sqrt{2}}+ y\left( \frac{1}{\sqrt{2}} \right) \right)^{2} equals :

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  3. Question 3

    4 Apr 2024

    180
    Let y=y(x)y=y(x) be the solution of the differential equation (x+y+2)2dx=dy,y(0)=−2(x+y+ 2)^{2}dx=dy,y(0) = - 2. Let the maximum and minimum values of the function y=y(x)y=y(x) in [0,π3]\left\lbrack 0,\frac{\pi}{3} \right\rbrack be α\alpha and β\beta, respectively. If (3α+π)2+β2=γ+δ3,γ,δ∈Z(3\alpha+\pi)^{2}+\beta^{2}=\gamma+\delta\sqrt{3},\gamma,\delta\in\mathbb{Z}, then γ+δ\gamma+\delta equals...........