Question ID: #1033
Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx} + 2y \sec^{2}x = 2\sec^{2}x + 3\tan x \sec^{2}x$ such that $y(0)=\frac{5}{4}$. Then $12\left(y\left(\frac{\pi}{4}\right)-e^{-2}\right)$ is equal to
Solution:
Standard form of a linear differential equation:
$$ \frac{dy}{dx} + Py = Q $$
Comparing with the given equation:
$$ P = 2\sec^2 x $$
$$ Q = 2\sec^2 x + 3\tan x \sec^2 x $$
Integrating Factor (IF):
$$ \text{IF} = e^{\int P dx} $$
$$ \text{IF} = e^{\int 2\sec^2 x dx} = e^{2\tan x} $$
The solution is given by the formula:
$$ y \cdot \text{IF} = \int Q \cdot \text{IF} dx + C $$
$$ y \cdot e^{2\tan x} = \int (2\sec^2 x + 3\tan x \sec^2 x) e^{2\tan x} dx $$
Let $\tan x = t$, which implies $\sec^2 x dx = dt$:
$$ y \cdot e^{2\tan x} = \int (2 + 3t) e^{2t} dt $$
Using integration by parts $\int u v dt = u \int v dt – \int \left( \frac{du}{dt} \int v dt \right) dt$:
$$ = (2+3t) \int e^{2t} dt – \int \left( \frac{d}{dt}(2+3t) \int e^{2t} dt \right) dt $$
$$ = (2+3t) \frac{e^{2t}}{2} – \int 3 \left(\frac{e^{2t}}{2}\right) dt $$
$$ = \frac{2+3t}{2} e^{2t} – \frac{3}{4} e^{2t} + C $$
$$ = e^{2t} \left( 1 + \frac{3}{2}t – \frac{3}{4} \right) + C $$
$$ = e^{2t} \left( \frac{1}{4} + \frac{3}{2}t \right) + C $$
Substitute $t = \tan x$ back into the equation:
$$ y \cdot e^{2\tan x} = e^{2\tan x} \left( \frac{1}{4} + \frac{3}{2}\tan x \right) + C $$
$$ y = \frac{1}{4} + \frac{3}{2}\tan x + C e^{-2\tan x} $$
Given the initial condition $y(0) = \frac{5}{4}$:
$$ \frac{5}{4} = \frac{1}{4} + \frac{3}{2}\tan(0) + C e^{-2\tan(0)} $$
$$ \frac{5}{4} = \frac{1}{4} + 0 + C(1) \Rightarrow C = 1 $$
The particular solution is:
$$ y = \frac{1}{4} + \frac{3}{2}\tan x + e^{-2\tan x} $$
Substitute $x = \frac{\pi}{4}$:
$$ y\left(\frac{\pi}{4}\right) = \frac{1}{4} + \frac{3}{2}\tan\left(\frac{\pi}{4}\right) + e^{-2\tan(\frac{\pi}{4})} $$
$$ y\left(\frac{\pi}{4}\right) = \frac{1}{4} + \frac{3}{2}(1) + e^{-2(1)} $$
$$ y\left(\frac{\pi}{4}\right) = \frac{7}{4} + e^{-2} $$
Calculate the required expression:
$$ 12\left(y\left(\frac{\pi}{4}\right) – e^{-2}\right) = 12\left(\frac{7}{4} + e^{-2} – e^{-2}\right) $$
$$ = 12 \left(\frac{7}{4}\right) = 21 $$
Ans. (21)
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