Question

Solve the differential equation: $\frac{dy}{dx}=x \sqrt{25-x^{2}}$

Answer 1

Solution:

first separate the variables then integrate the equation to find out the solution

$\frac{dy}{dx}=x \sqrt{25-x^{2}} \\ \\d y = x \sqrt{25-x^{2}} \: dx$
to integrate ,let us assume
$25 - {x}^{2} = t \\ \\ - 2x \: dx = dt \\ \\ xdx = \frac{dt}{ - 2} \\ \\ so \\ \\ \int1.dy =\int \frac{ - 1}{2} \sqrt{t} dt \\ \\ y = - \frac{1}{3} {t}^{ \frac{3}{2} } + c \\ \\ y = - \frac{1}{3} {(25 - {x}^{2} )}^{ \frac{3}{2} } + c$
is the solution.