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Question Number 169549 by ali009 last updated on 02/May/22

$$convertthisD.EtoexactD.Eandsolveit$$$$ydx+x(1+y)dy=0$$

Answered by aleks041103 last updated on 02/May/22

$$y+x(1+y)\frac{dy}{dx}=0$$$$\Rightarrow (1+y)\frac{dy}{dx}=-\frac{y}{x}$$$$\Rightarrow \frac{1+y}{y}dy=-\frac{dx}{x}$$$$\int \frac{1+y}{y}dy=y+ln\left(y\right)+C$$$$\int -\frac{dx}{x}=-ln\left(x\right)$$$$\Rightarrow y+ln\left(y\right)+ln\left(x\right)=C$$$${ye}^y=\frac{c}{x}$$$$\Rightarrow y\left(x\right)=W\left(\frac{c}{x}\right)$$

Commented by ali009 last updated on 02/May/22

$$welldone$$

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