Question

The differential equation is $\frac{dy}{dx} + y\tan x = \sec x$ and $y(0) = 1$. Then the value of $y(\frac{\pi}{4}) =$

• A. 0
• B. $\sqrt{2}$
• C. 1
• D. -1

Hint:

For linear equations $y' + Py = Q$, the integrating factor is always $e^{\int P dx}$. Here $P = \tan x$, making it easy!

Answer 1

The Correct Option is B

Step 1: Concept
Solve the linear differential equation using an Integrating Factor (I.F.).

Step 2: Meaning
$I.F. = e^{\int \tan x dx} = e^{\log \sec x} = \sec x$.

Step 3: Analysis
The general solution is $y(I.F.) = \int Q(I.F.) dx \implies y \sec x = \int \sec^2 x dx \implies y \sec x = \tan x + c$. Given $y(0)=1$: $(1)\sec(0) = \tan(0) + c \implies 1(1) = 0 + c \implies c = 1$.

Step 4: Conclusion
The solution is $y \sec x = \tan x + 1$. At $x = \pi/4$: $y \sec(\pi/4) = \tan(\pi/4) + 1 \implies y\sqrt{2} = 1 + 1 = 2 \implies y = 2/\sqrt{2} = \sqrt{2}$.
Final Answer: (B)