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  4. The solution of the differential equation (dy/dx)=(y/x)+(φ ( (y/x) )/φ ( (y)x )) is

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Q. The solution of the differential equation $ \frac{dy}{dx}=\frac{y}{x}+\frac{\phi \left( \frac{y}{x} \right)}{\phi \left( \frac{y}{x} \right)} $ is

KEAMKEAM 2007Differential Equations

Solution:

The given differential equation can be written as
$ \frac{dy}{dx}-\frac{y}{x}=\frac{\phi \left( \frac{y}{x} \right)}{\phi \left( \frac{y}{x} \right)} $
$ \Rightarrow $ $ x\phi \left( \frac{y}{x} \right)\left( \frac{1}{x}\frac{dy}{dx}-\frac{y}{{{x}^{2}}} \right)=\phi \left( \frac{y}{x} \right) $
$ \Rightarrow $ $ \frac{\phi \left( \frac{y}{x} \right)\left( \frac{x\frac{dy}{dx}-y}{{{x}^{2}}} \right)}{\phi \left( \frac{y}{x} \right)}=\frac{1}{x} $
$ \Rightarrow $ $ \int{\frac{\phi \left( \frac{y}{x} \right)d\left( \frac{y}{x} \right)}{\phi \left( \frac{y}{x} \right)}}=\int{\frac{1}{x}}dx+\log k $
$ \Rightarrow $ $ \log \phi \left( \frac{y}{x} \right)=\log x+\log k $
$ \Rightarrow $ $ \phi \left( \frac{y}{x} \right)=kx $