Question

Solve the initial value problem dydx=(x−2)(y−10),y(0)=5dydx=(x−2)(y−10),y(0)=5.

y=

Answer 1

The ODE is separable:

$\dfrac{\mathrm dy}{\mathrm dx}=(x-2)(y-10)\implies\dfrac{\mathrm dy}{y-10}=(x-2)\,\mathrm dx$

Integrating both sides gives

$\ln|y-10|=\dfrac{(x-2)^2}2+C$

$\implies y-10=e^{(x-2)^2/2+C}=Ce^{(x-2)^2/2}$

$\implies y=10+Ce^{(x-2)^2/2}$

Given that $y(0)=5$, we find

$5=10+Ce^{(-2)^2/2}\implies-5=Ce^2\implies C=-5e^{-2}$

so that

$\boxed{y(x)=10-5e^{(x-2)^2/2-2}}$