Answer to Question #196171 in Differential Equations for Ichigo

Question

Given dy/dx=y²/25 and y = 5 when x = 0 Find x when y =2

Answer 1

Let us solve the differential equation $\frac{dy}{dx}=\frac{y^2}{25}$:

$\frac{dy}{y^2}=\frac{dx}{25}$

$\int\frac{dy}{y^2}=\int\frac{dx}{25}$

$-\frac{1}{y}=\frac{x}{25}+C$

Since $y = 5$ when $x = 0$, we have that $C=-\frac{1}{5}.$

It follows that $-\frac{1}{y}=\frac{x}{25}-\frac{1}{5}.$

Let us find $x$ when $y =2:$

$-\frac{1}{2}=\frac{x}{25}-\frac{1}{5}$

$\frac{x}{25}=\frac{1}{5}-\frac{1}{2}=-\frac{3}{10}$

We conclude that $x=-\frac{15}{2}.$