Step 1: Understanding the Question:
The given equation \(\frac{dy}{dx} + y \tan x = \sec x\) is a first-order linear differential equation.
The general form is \(\frac{dy}{dx} + P(x)y = Q(x)\).
To solve this, we need to find the Integrating Factor (I.F.) and then the general solution.
Step 2: Key Formula or Approach:
Identify \(P(x) = \tan x\) and \(Q(x) = \sec x\).
Integrating Factor \(\text{I.F.} = e^{\int P(x) dx}\).
The general solution is \(y \cdot (\text{I.F.}) = \int [Q(x) \cdot (\text{I.F.})] dx + c\).
Step 3: Detailed Explanation:
Step A: Calculate Integrating Factor.
\[ \int P(x) dx = \int \tan x dx = \ln|\sec x| \]
\[ \text{I.F.} = e^{\ln|\sec x|} = \sec x \]
Step B: Find General Solution.
\[ y \cdot \sec x = \int (\sec x \cdot \sec x) dx \]
\[ y \sec x = \int \sec^2 x dx \]
\[ y \sec x = \tan x + c \] ... (1)
Step C: Apply Initial Condition \(y(0) = 1\).
\[ 1 \cdot \sec(0) = \tan(0) + c \]
\[ 1 \cdot 1 = 0 + c \implies c = 1 \]
Step D: The particular solution is \(y \sec x = \tan x + 1\).
Rewriting for \(y\): \(y = \frac{\tan x + 1}{\sec x} = \frac{\frac{\sin x}{\cos x} + 1}{\frac{1}{\cos x}} = \sin x + \cos x\).
Step E: Evaluate at \(x = \frac{\pi}{4}\).
\[ y\left(\frac{\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) + \cos\left(\frac{\pi}{4}\right) \]
\[ y\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \]
Step 4: Final Answer:
The solution simplifies to \(y = \sin x + \cos x\). At \(\pi/4\), both terms are \(1/\sqrt{2}\), making the sum \(\sqrt{2}\).