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  4. The general solution of the differential equation (d y/d x)+y g prime(x)= g (x) ⋅ g prime(x) where g(x) is a given function of x, is

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Q. The general solution of the differential equation $\frac{d y}{d x}+y g^{\prime}(x)= g (x) \cdot g^{\prime}(x)$ where $g(x)$ is a given function of $x$, is

Differential Equations

Solution:

We have,
$\frac{d y}{d x}=(g(x)-y) \cdot g^{\prime}(x) $
Put $ g(x)-y=V $
$\Rightarrow g^{\prime}(x)-\frac{d y}{d x}=\frac{d V}{d x}$
Hence $ g^{\prime}(x)-\frac{d V}{d x}=V \cdot g^{\prime}(x) $
$\Rightarrow \frac{d V}{d x}=(1-V) g^{\prime}(x) $
$\Rightarrow \frac{d V}{1-V}=g^{\prime}(x) d x $
$\Rightarrow \int \frac{d V}{1-V}=\int g^{\prime}(x) d x$
$\Rightarrow -\log (1-V)=g(x)-c$
$\Rightarrow g(x)+\log (1-V)=c $
$\therefore g(x)+\log [1+y-g(x)]=c$