23.2 Resolution

E426

[1R1]Let \(Θ:ℝ→ℝ\) be a continuous function, Describe all solutions \(f:ℝ→ℝ\) that solve

\[ ∀ x≠ 0~ ,~ f'(x) = Θ\left(\frac{f(x)}{x}\right) \]

(Hint: change variables \(f(x) = x h(x)\) and find and solve a differential equation for \(h(x)\).)

Hidden solution: [UNACCESSIBLE UUID ’1R2’]

[UNACCESSIBLE UUID ’1R3’]

[1R4]Find solutions to the problem

\[ \frac{dy}{dx}=\frac{y}{x+y} \]

with substitution \(z=y/x\), and also comparing it with the problem

\[ \frac{dx}{dy}=\frac{x+y}{y}~ . \]

[UNACCESSIBLE UUID ’1R5’]