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:

Step 1: Identify the form of the differential equation and find the integrating factor

The given differential equation is of the form of a linear first-order differential equation, which is $\frac{dy}{dx} + P(x)y = Q(x)$. Here, $P(x) = 2\sec^2 x$ and $Q(x) = 2\sec^2 x + 3\tan x \cdot \sec^2 x$. To solve this, we first need to find the integrating factor (I.F.). The integrating factor is given by the formula $e^{\int P(x) dx}$.

Step 2: Calculate the integrating factor

We substitute $P(x) = 2\sec^2 x$ into the integrating factor formula. The integral of $\sec^2 x$ is $\tan x$. Therefore, the integrating factor is $e^{2\tan x}$.

Step 3: Solve the differential equation

The general solution for a linear first-order differential equation is given by $y \cdot (I.F.) = \int Q(x) \cdot (I.F.) dx$. We substitute the expressions for $Q(x)$ and the integrating factor into this formula. To solve the integral on the right-hand side, we can use a substitution.

Step 4: Evaluate the integral using substitution

Let $t = 2\tan x$, so $dt = 2\sec^2 x dx$. The integral becomes $\int (1 + \frac{3}{2}t) e^t \frac{dt}{2}$. We can split this into two integrals and use integration by parts for the term with $t e^t$. After integrating and substituting back $t = 2\tan x$, we simplify the expression to find the general solution for $y$.

Step 5: Apply the initial condition to find C

We are given the initial condition $y(0) = \frac{5}{4}$. We substitute $x=0$ and $y=\frac{5}{4}$ into the general solution. Since $\tan(0) = 0$ and $e^0 = 1$, we can solve for the constant $C$.

Step 6: Find $y\left(\frac{\pi}{4}\right)$

Now that we have the value of $C$, we can write the particular solution. We need to find $y\left(\frac{\pi}{4}\right)$. We substitute $x = \frac{\pi}{4}$ into the particular solution. Since $\tan\left(\frac{\pi}{4}\right) = 1$, we can simplify the expression.

Step 7: Calculate the final expression

Finally, we need to calculate the value of $12 \left( y\left(\frac{\pi}{4}\right) - e^{-2} \right)$. We substitute the value of $y\left(\frac{\pi}{4}\right)$ we just found and simplify the expression.

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