Differential Equations
Homogeneous ODE — Implicit Solution
nta_pyq_2023_jan
Grade None
Question:
The solution of the differential equation $\dfrac{dy}{dx}=-\left(\dfrac{x^2+3y^2}{3x^2+y^2}\right)$, $y(1)=0$ is:
$\log_e|x+y|-\dfrac{xy}{(x+y)^2}=0$
$\log_e|x+y|+\dfrac{xy}{(x+y)^2}=0$
$\log_e|x+y|+\dfrac{2xy}{(x+y)^2}=0$
$\log_e|x+y|-\dfrac{2xy}{(x+y)^2}=0$
Step-by-Step Solution
Key Concept: Homogeneous: $v=y/x$. $v\,dv=-dx/(3x)$. Integrate: $v^2/2=-\ln x/3+C$. Back-substitute.
$\log_e|x+y|+\dfrac{2xy}{(x+y)^2}=0$.
Correct Answer: 3