Let f be a differentiable function such that 2(x+2)^2f(x)-3(x+2)^2=10 _x^0(t+2)f(t)dt,x 0.

Question

Let $f$ be a differentiable function such that $2(x+2{)}^{2}f(x)-3(x+2{)}^{2}=10{\int }_{x}^{0}(t+2)f(t)dt,x\ge 0$. Then $f(2)$ is equal to \_\_\_\_\_\_\_\_ .

Answer: 19

Accepted Answer

$\begin{gathered} \ 4(x+2)f(x)+2(x+2{)}^{2}{f}^{'}(x)-6(x+2)=10(x+2)f(x) \ 2(x+2{)}^{2}{f}^{'}(x)-6(x+2)f(x)=6(x+2) \ (x+2)\frac{dy}{dx}-3y=3 \ \int \frac{dy}{dx}=3\int \frac{dx}{x+2} \ \ln (y+1)=3\ln (x+2)+C \ (y+1)=C(x+2{)}^{3} \ f(0)=\frac{3}{2} \ f(2)=19 \end{gathered}$

Solutions · 1