Find the general solution of $x \frac{d y}{d x} = x y + y$ $xdy/dx = xy + y$ ?

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Answer 1 · 2018-06-10T12:19:50

$y = C_{1} x e^{x}$ $y=C_1xe^x$

We want to solve the differential equation

$x \frac{d y}{d x} = x y + y$ $xdy/dx=xy+y$

This can solved by separation of variables

Separate the variables

$x \frac{d y}{d x} = y (x + 1)$ $xdy/dx=y(x+1)$

$\Rightarrow \frac{1}{y} d y = \frac{x + 1}{x} d x$ $=>1/ydy=(x+1)/xdx$

Integrate both sides

$\int \frac{1}{y} d y = \int \frac{x + 1}{x} d x$ $int1/ydy=int(x+1)/xdx$

$\Rightarrow ln (y) = x + ln (x) + C$ $=>ln(y)=x+ln(x)+C$

Solve for $y$ $y$

$y = e^{x + ln (x) + C}$ $y=e^(x+ln(x)+C)$

Simplify

$y = C_{1} x e^{x}$ $y=C_1xe^x$

Find the general solution of xdydx=xy+yxdy/dx = xy + y ?