If x=x(t) solves (t+1),dx=(2x+(t+1)^4),dt, x(0)=2, then x(1) equals ______.

Question

If $x=x(t)$ solves $(t+1)\,dx=(2x+(t+1)^4)\,dt$, $x(0)=2$, then $x(1)$ equals \_\_\_\_\_\_.

Answer: 14

Accepted Answer

$$\frac{dx}{dt}-\frac{2}{t+1}x = (t+1)^3.$$

I.F. $=(t+1)^{-2}$.
$$\frac{x}{(t+1)^2} = \int(t+1)\,dt = \frac{(t+1)^2}{2}+C.$$

Using $x(0)=2$: $2=\dfrac12+C \Rightarrow C=\dfrac32$.

At $t=1$: $\dfrac{x}{4}=2+\dfrac32=\dfrac72 \Rightarrow x=14$.

Solutions · 1