Differential Equations
Bernoulli ODE — Exponential Relation
nta_pyq_2023_apr
Grade 12
Question:
Let $y=y(x)$, $y>0$, be a solution of $(1+x^2)\,dy=y(x-y)\,dx$ with $y(0)=1$ and $y(2\sqrt{2})=\beta$. Then
$e^{3\beta-1}=e(3+2\sqrt{2})$
$e^{3\beta-1}=e(5+\sqrt{2})$
$e^{\beta-1}=e^{-2(3+2\sqrt{2})}$
$e^{\beta-1}=e^{-2(5+\sqrt{2})}$
Step-by-Step Solution
Key Concept: Substitute $\frac{1}{y}=t$. Bernoulli: $-\frac{1}{y^2}y'+\frac{x}{1+x^2}\cdot\frac{1}{y}=\frac{1}{1+x^2}$. Linear ODE in $t$.
$\frac{3}{\beta}=1+\ln(3+2\sqrt{2})$. Rearranging to match option (1): $e^{3\beta-1}=e(3+2\sqrt{2})$.
Correct Answer: 1