Differential Equations
Linear ODE — Product αβγ from Curve Parameters
nta_pyq_2023_apr
Grade 12
Question:
If $y=y(x)$ is the solution of $\dfrac{dy}{dx}+\dfrac{4x}{x^2-1}y=\dfrac{x+2}{5(x^2-1)^2}$, $x>1$, with $y(2)=\dfrac{2}{9}\ln_e(2+\sqrt{3})$, and $y(\sqrt{2})=\alpha\ln_e(\sqrt{\alpha}+\beta)+\beta-\sqrt{\gamma}$, then $\alpha\beta\gamma$ is equal to
Step-by-Step Solution
Key Concept: IF $=(x^2-1)^2$. Solution: $y(x^2-1)^2=2\ln|x+\sqrt{x^2-1}|+\sqrt{x^2-1}+C$. Use $y(2)$ to find $C=-\sqrt{3}$.
$\alpha=2,\beta=1,\gamma=3$. $\alpha\beta\gamma=6$.
Correct Answer: 6