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

Question

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

Dr. Abubacker · Accepted Answer

[Solution by Dr. Abubacker] $f'-\frac2xf=\frac3{x^2}$. With integrating factor $x^{-2}$: $\left(\frac f{x^2}\right)'=\frac3{x^4}\Rightarrow\frac f{x^2}=-\frac1{x^3}+C$. $f(1)=4\Rightarrow C=5$, so $f(x)=5x^2-\frac1x$. $f(2)=20-\frac12=19.5$, hence $2f(2)=39$. Answer (1).

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

Explanation

Reviewed by Dr. Abubacker

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