Differential Equations
Separable Equation with $\tan^{-1}$
nta_pyq_2024_apr
Grade 12
Question:
Let $y=y(x)$ be the solution of the differential equation $(1+y^2)e^{\tan x}\,dx+\cos^2x(1+e^{2\tan x})\,dy=0$, $y(0)=1$. Then $y\left(\dfrac{\pi}{4}\right)$ is equal to:
$\dfrac{2}{e}$
$\dfrac{2}{e^2}$
$\dfrac{1}{e}$
$\dfrac{1}{e^2}$
Step-by-Step Solution
Key Concept: Separate: $\frac{\sec^2x\cdot e^{\tan x}}{1+e^{2\tan x}}dx+\frac{dy}{1+y^2}=0$. Integrate: $\tan^{-1}(e^{\tan x})+\tan^{-1}y=C$.
$y(\pi/4)=1/e$.
Correct Answer: 3