Let f:(0,) R be a function which is differentiable at all points of its domain and satisfi…

Question

Let $f:(0,\infty)	o R$ be a function which is differentiable at all points of its domain and satisfies the condition ${x}^{2}{f}^{'}(x)=2xf(x)+3$, with $f(1)=4$. Then $2f(2)$ is equal to :

Accepted Answer

$\begin{gathered} {x}^{2}{f}^{'}(x)-2xf(x)=3 \ (\frac{{x}^{2}{f}^{'}(x)-2xf(x)}{{({x}^{2})}^{2}})=\frac{3}{{({x}^{2})}^{2}} \ \Rightarrow \frac{d}{dx}(\frac{f(x)}{{x}^{2}})=\frac{3}{{x}^{4}} \end{gathered}$

Integrating both sides

$\begin{gathered} \frac{f(x)}{{x}^{2}}=-\frac{1}{{x}^{3}}+C \ f(x)=-\frac{1}{x}+{Cx}^{2} \ x=1 \ 4=-1+C\Rightarrow C=5 \ f(x)=-\frac{1}{x}+5{x}^{2} \end{gathered}$

Now $2	imes f(2)=2	imes [-\frac{1}{2}+5	imes {2}^{2}]$

$=39$

Solutions · 1