Limits, Continuity & Differentiability
Differential Equation from Functional Definition
nta_pyq_2024_jan
Grade 12
Question:
Let $f(x)=\sqrt{\displaystyle\lim_{r\to x}\left\{\frac{2r^2[(f(r))^2-f(x)f(r)]}{r^2-x^2}-r^3 e^{f(r)/r}\right\}}$ be differentiable in $(-\infty,0)\cup(0,\infty)$ and $f(1)=1$. Then the value of $ea$, such that $f(a)=0$, is equal to ______.
Step-by-Step Solution
Key Concept: Evaluate the limit as $r\to x$: the expression inside simplifies to $2x^2f(x)f'(x)/(2x)-x^3e^{f(x)/x}=xf(x)f'(x)-x^3e^{f(x)/x}$. So $f^2(x)=xf(x)f'(x)-x^3e^{f(x)/x}$, giving a homogeneous ODE. Substitute $y=f(x)$, $v=y/x$.
Simplifying the limit gives the ODE: $y^2=xy\frac{dy}{dx}-x^3e^{y/x}$. Divide by $x^2$: $v^2=v(v+x\frac{dv}{dx})-xe^v$, i.e. $x\frac{dv}{dx}=-xe^v/v$... After full simplification and integration with $f(1)=1$: setting $f(a)=0$ gives $ae=2$.
Correct Answer: 2