Differential Equations

Question

Let $f(x)$ be differentiable s.t. $f'(x) = 2 - f(x)/x$ and $f(1)=1$. Then $f(x)$ is:

RankGuru · Accepted Answer

1. Rearrange to standard Linear DE form:
$f'(x) + \frac{1}{x}f(x) = 2$.
$P(x) = 1/x, Q(x) = 2$.

2. Integrating Factor:
IF $= e^{\int 1/x dx} = x$.

3. Solve:
$x f(x) = \int 2x dx + C$
$x f(x) = x^2 + C$
$f(x) = x + C/x$.

4. Apply condition $f(1) = 1$:
$1 = 1 + C/1 \implies C = 0$.
So, $f(x) = x$.

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