Differential Equations
Homogeneous ODE — Bernoulli or Substitution
nta_pyq_2023_apr
Grade 12
Question:
The slope of tangent at any point $(x,y)$ on a curve $y=y(x)$ is $\dfrac{x^2+y^2}{2xy}$, $x>0$. If $y(2)=0$, then a value of $y(8)$ is
-4\sqrt{2}
2\sqrt{3}
-2\sqrt{3}
4\sqrt{3}
Step-by-Step Solution
Key Concept: Homogeneous ODE: put $y=vx$. $\frac{dv}{dx}=\frac{1-v^2}{2vx}\Rightarrow\int\frac{2v}{v^2-1}dv=-\int\frac{dx}{x}$. $\ln|v^2-1|=-\ln x+C$.
$y^2=x^2-2x$. $y(8)=4\sqrt{3}$.
Correct Answer: 4