Solve the following differential equations : (x+a) d y d x =- y +… - Vidyadip

Question

Solve the following differential equations : $(x+a) \frac{d y}{d x}=- y + a$

✓

Answer

$
\begin{aligned}
& ( x + a ) \frac{d y}{d x}+ y = a \\
& \therefore \frac{d y}{d x}+\left(\frac{1}{x+a}\right) y=\frac{a}{x+a}
\end{aligned}
$
This is the linear differential equation of the form
$
\frac{d y}{d x}+P \cdot y=Q \text {, where } P=\frac{1}{x+a}, Q=\frac{a}{x+a} \text {. }
$
$
\begin{aligned}
\text { I.F. } & =e^{\int P d x}=e^{\int \frac{1}{x+a} d x} \\
& =e^{\log (x+a)}=x+a
\end{aligned}
$
$\therefore$ the solution of (1) is given by
$
\begin{aligned}
y \cdot(\text { I.F. }) & =\int Q \cdot \text { (I.F.) } d x+c \\
\therefore y(x+a) & =\int\left(\frac{a}{x+a}\right)(x+a) d x+c \\
& =a \int d x+c \\
\therefore y(x+a) & =a x+c
\end{aligned}
$
This is the general solution.

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