The solution of the differential equation dydx=2xy−3y+2x−3 is

dydx=2xy−3y+2x−3=(y+1)(2x−3)dx∫dyy+1=∫(2x−3)dxln⁡|y+1|+ln⁡c=x2−3x⇒ln⁡|c(y+1)|=x2−3x⇒c(y+1)=ex2−3x

The solution of the differential equation dydx=2xy−3y+2x−3 is

dydx=2xy−3y+2x−3=(y+1)(2x−3)dx∫dyy+1=∫(2x−3)dxln⁡|y+1|+ln⁡c=x2−3x⇒ln⁡|c(y+1)|=x2−3x⇒c(y+1)=ex2−3x

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The solution of the differential equation $\frac{d y}{d x} = 2 x y - 3 y + 2 x - 3$ is

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$c (y + 1) = e^{x^{2} - 3 x}$

$c x (y + 1) = e^{x^{2}}$

$c y (x + 1) = e^{y^{2}}$

$e^{y^{2} - 3 y} = 3 x + c$

answer is A.

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Detailed Solution

$\begin{array}{l} \frac{d y}{d x} = 2 x y - 3 y + 2 x - 3 = (y + 1) (2 x - 3) d x \\ \int \frac{d y}{y + 1} = \int (2 x - 3) d x \\ \ln | y + 1 | + \ln c = x^{2} - 3 x \\ \Rightarrow \ln | c (y + 1) | = x^{2} - 3 x \\ \Rightarrow c (y + 1) = e^{x^{2} - 3 x} \end{array}$