Question

Solve the differential equation.
(x + y) dx – x dy = 0​

Answer 1

Given,

$(x+y)dx-x\ dy=0\quad\longrightarrow\quad (1)$

Well, solving it means we have to express $y$ in terms of $x.$ So let,

$y=ux\quad\longrightarrow\quad (2)$

where $u$ is a function in $x,$ not a constant, beware.

Then,

$\dfrac {dy}{dx}=\dfrac {d}{dx}(ux)\\\\\\\dfrac {dy}{dx}=u\dfrac {dx}{dx}+x\dfrac {du}{dx}\\\\\\\dfrac {dy}{dx}=u+x\dfrac {du}{dx}\\\\\\dy=u\ dx+x\ du$

Then (1) becomes,

$(x+ux)dx-x(u\ dx+x\ du)=0\\\\\\x\ dx+ux\ dx-ux\ dx-x^2\ du=0\\\\\\x\ dx=x^2\ du\\\\\\du=\dfrac {1}{x}\ dx$

Now, it's time for integration!

$\displaystyle\int du=\int\dfrac {1}{x}\ dx\\\\\\u=\ln|x|+c$

where $c$ is the integral constant.

Now, from (2),

$\boxed {\boxed {\mathbf {y=x\ln|x|+cx}}}$

#answerwithquality

#BAL