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QJEEM25Ma4e26: JEE Main 2025 Mathematics - Class 12 JEEM25Ma4e26 Solution: Let be a differentiable function such that . Then is equal to…

Let ff be a differentiable function such that 2(x+2)2f(x)−3(x+2)2=10∫0x(t+2)f(t)dt,x≥02(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)f(2) is equal to ________ .

Step-by-Step Verified Solution

amp; Differentiate both sides amp;amp;4(x+2)f(x)+2(x+2)2f′(x)−6(x+2)=10(x+2)f(x)amp;2(x+2)2f′(x)−6(x+2)f(x)=6(x+2)amp;(x+2)dydx−3y=3amp;∫dydx=3∫dxx+2amp;ln⁡(y+1)=3ln⁡(x+2)+Camp;(y+1)=C(x+2)3amp;f(0)=32amp;f(2)=19\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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