Differential Equations
Linear ODE / Integrating Factor
MJMT_Full_Test_10
Grade 12
Question:
Let $y=y(x)$ be a solution of $\dfrac{dy}{dx} + \dfrac{y\cos x}{\sin x+\cos x} = \dfrac{2\tan^2 x+\tan x+2}{\sqrt{\sin x+\cos x}}$. If $y\!\left(\dfrac{\pi}{4}\right)=2^{3/4}$, then $y^2\!\left(\dfrac{7\pi}{12}\right)$ is
Step-by-Step Solution
Key Concept: Integrating factor $=e^{\int\frac{\cos x}{\sin x+\cos x}dx}$; compute the integral of the form $\frac{1}{2}(x+\ln|\sin x+\cos x|)$.
IF $=e^{\frac{1}{2}(x+\ln|\sin x+\cos x|)}=(\sin x+\cos x)^{1/2}e^{x/2}$. Multiply and integrate RHS. After integration and applying IC $y(\pi/4)=2^{3/4}$, find $y(7\pi/12)^2=28\sqrt{2}+16\sqrt{6}$.
Correct Answer: 4