Let $f$ be a differentiable function such that $2(x+2)^2 f(x)-3(x+2)^2=10 \int_0^x(t+2) f(t) d t, x \geq 0$. Then $f(2)$ is equal to ______.

Question

Aicharya · Accepted Answer

$2(x+2)^2 f(x)-3(x+2)^2=10 \int_0^x(t+2) f(t) d t$
Differentiating both side
$\begin{aligned}
& 4(x+2) f(x)+2 f(x)(x+2)^2-6(x+2)=10(x+2) f(x) \\ & =(x+2) \frac{d y}{d x}-3 y=3 \\ & \frac{1}{3} \int \frac{d y}{y+1}=\int \frac{d x}{x+2} \\ & \ln |y+1=3 \ln | x+2 \mid+\ln c \\ & y+1=(x+2)^3 c \\ & \because y(0)=\frac{3}{2} \\ & \Rightarrow \frac{5}{16}=c \\ & \therefore \quad y=\frac{5}{16}(x+2)^3-1 \\ & y(2)=\frac{5}{16} \times 64-1=19
\end{aligned}$

Let $f$ be a differentiable function such that $2(x+2)^2 f(x)-3(x+2)^2=10 \int_0^x(t+2) f(t) d t, x \geq 0$.…

Solution