Solution of Second Order Non Linear Differential Equation Example

To solve the differential equation  
\[y \frac{d^2y}{dx^2} + (\frac{dy}{dx})^2=1\]
 let  
\[\frac{dy}{dx}=p\]
  then  
\[\frac{d^2y}{dx^2}= \frac{dp}{dx}= \frac{dy}{dx} \frac{dp}{dy}=p \frac{dy}{dx}\]
  and the equation becomes  
\[yp \frac{dp}{dy}+p^2=1\]
.
Separation of variables gives  
\[\frac{p}{1-p^2}dp= \frac{1}{y} dy\]

\[\int \frac{p}{1-p^2}dp= \int \frac{1}{y} dy\]

\[- \frac{1}{2} ln(1-p^2)= lny+c= \]

\[ln(1-p^2)=-2lny-2c=ln(\frac{1}{y^2})-2c\]

\[1-p^2=e^{ln(\frac{1}{y^2})-2c}=e^{-2c} \frac{1}{y^2}\]

\[p=\sqrt{1-e^{-2c} \frac{1}{y^2}}=\sqrt{y^2-A}{y}\]

\[p= \frac{dy}{dx}\]
  so
\[\frac{dy}{dx}=\sqrt{y^2-A}{y}\]

Separating variables again  
\[\frac{y}{\sqrt{y^2-A}}dy= dx\]
.
Integration gives  
\[\sqrt{y^2-A}=x+B \rightarrow y^2=(x+B)^2+A \rightarrow y=\sqrt{(x+B)^2+A}\]
.

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