Question

Let \( y = y(x) \) be the solution of the differential equation \[ \frac{dy}{dx} + 2y \sec^2 x = 2 \sec^2 x + 3 \tan x \cdot \sec^2 x \] such that \( y(0) = \frac{5}{4} \). Then \[ 12 \left( y\left( \frac{\pi}{4} \right) - e^{-2} \right) \] is equal to _____.

Hint:

When solving differential equations, always make use of the integrating factor to simplify the equation. Ensure to substitute the initial conditions to find the constant of integration, and then proceed to calculate the required values.

Answer 1

Correct Answer: 21

Let \( y=y(x) \) solve the linear differential equation \( \dfrac{dy}{dx} + 2y\sec^2 x = 2\sec^2 x + 3\tan x \cdot \sec^2 x \) with \( y(0)=\dfrac{5}{4} \). We are to evaluate \( 12\!\left( y\!\left(\dfrac{\pi}{4}\right) - e^{-2}\right) \).

Concept Used:

This is a first-order linear ODE \( y' + P(x)y = Q(x) \). The integrating factor is \( \mu(x)=e^{\int P(x)\,dx} \). The solution is \( y\mu = \int \mu Q\,dx + C \).

\[ P(x)=2\sec^2 x \;\Rightarrow\; \mu(x)=e^{\int 2\sec^2 x\,dx}=e^{2\tan x}. \]

Step-by-Step Solution:

Step 1: Multiply the ODE by the integrating factor \(e^{2\tan x}\) to get an exact derivative:

\[ \frac{d}{dx}\!\Big( y\,e^{2\tan x} \Big) \;=\; e^{2\tan x}\!\left(2\sec^2 x + 3\tan x\,\sec^2 x\right). \]

Step 2: Integrate the right-hand side using the substitution \(u=\tan x,\; du=\sec^2 x\,dx\):

\[ \int e^{2\tan x}\!\left(2\sec^2 x + 3\tan x\,\sec^2 x\right)\!dx = \int e^{2u}(2+3u)\,du. \] \[ \int e^{2u}(2+3u)\,du = e^{2u}\!\left(\frac{3u}{2}+\frac{1}{4}\right)+C. \]

Step 3: Return to \(x\) and solve for \(y\):

\[ y\,e^{2\tan x}=e^{2\tan x}\!\left(\frac{3}{2}\tan x+\frac{1}{4}\right)+C \;\Rightarrow\; y=\frac{3}{2}\tan x+\frac{1}{4}+C\,e^{-2\tan x}. \]

Step 4: Use the initial condition \(y(0)=\dfrac{5}{4}\) (\(\tan 0=0\)) to find \(C\):

\[ \frac{5}{4}=0+\frac{1}{4}+C \;\Rightarrow\; C=1. \] \[ \therefore\; y(x)=\frac{3}{2}\tan x+\frac{1}{4}+e^{-2\tan x}. \]

Final Computation & Result

At \(x=\dfrac{\pi}{4}\), \(\tan\dfrac{\pi}{4}=1\), hence

\[ y\!\left(\frac{\pi}{4}\right)=\frac{3}{2}\cdot 1+\frac{1}{4}+e^{-2}=\frac{7}{4}+e^{-2}. \] \[ 12\left( y\!\left(\frac{\pi}{4}\right)-e^{-2}\right)=12\left(\frac{7}{4}\right)=21. \]

Answer: 21