The solution of dydx=x+x2y+y2 is

dydx=x+x2y+y2⇒y+y2dy=x+x2dx⇒∫y+y2dy=∫x+x2dx⇒y22+y33=x22+x33−c6⇒2x3−y3+3x2−y2=c

The solution of dydx=x+x2y+y2 is

dydx=x+x2y+y2⇒y+y2dy=x+x2dx⇒∫y+y2dy=∫x+x2dx⇒y22+y33=x22+x33−c6⇒2x3−y3+3x2−y2=c

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The solution of $\frac{d y}{d x} = \frac{x + x^{2}}{y + y^{2}}$ is

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$\begin{aligned} 2 (x^{3} - y^{3}) + 3 (x^{2} - y^{2}) = c \end{aligned}$

$\begin{aligned} 2 (x^{3} - y^{3}) - 3 (x^{2} - y^{2}) = c \end{aligned}$

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

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

answer is A.

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Detailed Solution

$\begin{array}{l} \frac{d y}{d x} = \frac{x + x^{2}}{y + y^{2}} \Rightarrow (y + y^{2}) d y = (x + x^{2}) d x \Rightarrow \int (y + y^{2}) d y = \int (x + x^{2}) d x \Rightarrow \frac{y^{2}}{2} + \frac{y^{3}}{3} = \frac{x^{2}}{2} + \frac{x^{3}}{3} - \frac{c}{6} \\ \Rightarrow 2 (x^{3} - y^{3}) + 3 (x^{2} - y^{2}) = c \end{array}$