Mathematics - Differential Equation Question with Solution | TestHub

$2(x+2{)}^{2}f(x)-3(x+2{)}^2=10{\int }_0^x(t+2)f(t)dt$ Differentiating both side $\begin{array}{rl}& 4(x+2)f(x)+2f(x)(x+2{)}^2-6(x+2)=10(x+2)f(x)\\ & =(x+2)\frac{dy}{dx}-3y=3\\ & \frac{1}{3}\int \frac{dy}{y+1}=\int \frac{dx}{x+2}\\ & \ln |y+1=3\ln |x+2\mid +\ln c\\ & y+1=(x+2{)}^{3}c\\ & \because y(0)=\frac{3}{2}\\ & \Rightarrow \frac{5}{16}=c\\ & \therefore y=\frac{5}{16}(x+2{)}^3-1\\ & y(2)=\frac{5}{16}\times 64-1=19\end{array}$