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:

`(\mathrm dy)/(\mathrm dx)=(x-2)(y-10)\implies(\mathrm dy)/(y-10)=(x-2)\,\mathrm dx`

Integrating both sides gives

`\ln|y-10|=\frac{(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)}`