Differential Equations
Exact Equation
nta_pyq_2024_apr
Grade 12
Question:
If the solution $y(x)$ of the given differential equation $(e^y+1)\cos x\,dx+e^y\sin x\,dy=0$ passes through the point $\left(\dfrac{\pi}{2},0\right)$, then the value of $e^{y(\pi/6)}$ is equal to ________.
Step-by-Step Solution
Key Concept: Recognize: $d[(e^y+1)\sin x]=\cos x(e^y+1)dx+e^y\sin x\,dy=0$. So $(e^y+1)\sin x=C$.
$(e^y+1)\sin x=2$. At $x=\pi/6$: $e^y=3$.
Correct Answer: 3