If `dy/dx+3/cos^2xy=1/cos^2x,x in((-pi)/3,pi/3)and y(pi/4)=4/3," then "y(-pi/4)` equals

Options

A. `1/3+e^6`
B. `1/3`
C. `-1/4`
D. `1/3+e^3`

Answer

To solve the differential equation given in the problem, we will follow a systematic approach.

### Step-by-Step Solution:

1. **Write the Differential Equation**:
The given equation is:
\[ \frac{dy}{dx} + \frac{3}{\cos^2 x} y = \frac{1}{\cos^2 x} \]

2. **Identify p(x) and q(x)**:
Here, we can identify:
\[ p(x) = \frac{3}{\cos^2 x}, \quad q(x) = \frac{1}{\cos^2 x} \]

3. **Find the Integrating Factor**:
The integrating factor \( \mu(x) \) is given by:
\[ \mu(x) = e^{\int p(x) \, dx} = e^{\int \frac{3}{\cos^2 x} \, dx} \]
The integral of \( \frac{3}{\cos^2 x} \) is:
\[ \int \frac{3}{\cos^2 x} \, dx = 3 \tan x \]
Therefore, the integrating factor is:
\[ \mu(x) = e^{3 \tan x} \]

4. **Multiply the Equation by the Integrating Factor**:
Multiply the entire differential equation by \( e^{3 \tan x} \):
\[ e^{3 \tan x} \frac{dy}{dx} + 3 e^{3 \tan x} \frac{y}{\cos^2 x} = e^{3 \tan x} \frac{1}{\cos^2 x} \]

5. **Rewrite the Left Side**:
The left side can be rewritten as:
\[ \frac{d}{dx} \left( e^{3 \tan x} y \right) = e^{3 \tan x} \sec^2 x \]

6. **Integrate Both Sides**:
Integrate both sides:
\[ e^{3 \tan x} y = \int e^{3 \tan x} \sec^2 x \, dx \]
The right side can be solved using substitution. Let \( u = \tan x \), then \( du = \sec^2 x \, dx \):
\[ \int e^{3u} \, du = \frac{1}{3} e^{3u} + C = \frac{1}{3} e^{3 \tan x} + C \]

7. **Solve for y**:
Thus, we have:
\[ e^{3 \tan x} y = \frac{1}{3} e^{3 \tan x} + C \]
Dividing by \( e^{3 \tan x} \):
\[ y = \frac{1}{3} + C e^{-3 \tan x} \]

8. **Use Initial Condition to Find C**:
We know \( y\left(\frac{\pi}{4}\right) = \frac{4}{3} \):
\[ \frac{4}{3} = \frac{1}{3} + C e^{-3 \cdot 1} \]
Simplifying gives:
\[ C e^{-3} = \frac{4}{3} - \frac{1}{3} = 1 \implies C = e^{3} \]

9. **Final Solution for y**:
The solution for \( y \) is:
\[ y = \frac{1}{3} + e^{3} e^{-3 \tan x} \]

10. **Find y(-π/4)**:
Now, we need to find \( y\left(-\frac{\pi}{4}\right) \):
\[ y\left(-\frac{\pi}{4}\right) = \frac{1}{3} + e^{3} e^{-3 \cdot (-1)} = \frac{1}{3} + e^{3} e^{3} = \frac{1}{3} + e^{6} \]

### Final Answer:
\[ y\left(-\frac{\pi}{4}\right) = \frac{1}{3} + e^{6} \]