JEE MAIN - Mathematics (2025 - 24th January Morning Shift - No. 22)
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 ________ .
Answer
19
Explanation
$$\begin{aligned}
&\text { Differentiate both sides }\\
&\begin{aligned}
& 4(x+2) f(x)+2(x+2)^2 f^{\prime}(x)-6(x+2)=10(x+2) f(x) \\
& 2(x+2)^2 f^{\prime}(x)-6(x+2) f(x)=6(x+2) \\
& (x+2) \frac{d y}{d x}-3 y=3 \\
& \int \frac{d y}{d x}=3 \int \frac{d x}{x+2} \\
& \ln (y+1)=3 \ln (x+2)+C \\
& (y+1)=C(x+2)^3 \\
& f(0)=\frac{3}{2} \\
& f(2)=19
\end{aligned}
\end{aligned}$$
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