Question

If $\frac{dy}{dx} +2y = sin 2x$ and $y(0) = \frac{3}4$ , then $y (\frac{π}{8})$ is equal to:

• A. $e^{\frac{\pi}{8}}$
• B. $e^{\frac{\pi}{6}}$
• C. $e^{\frac{-\pi}{4}}$
• D. None

Answer 1

The Correct Option is C

To solve the differential equation \(\frac{dy}{dx} + 2y = \sin 2x\) with the initial condition \(y(0) = \frac{3}{4}\) and to find \(y\left(\frac{\pi}{8}\right)\), we can use the method of integrating factors. 

1. The given differential equation is in the standard form \(\frac{dy}{dx} + P(x)y = Q(x)\), where \(P(x) = 2\) and \(Q(x) = \sin 2x\).
2. The integrating factor, \(\mu(x)\), is given by \(\mu(x) = e^{\int P(x) \, dx} = e^{\int 2 \, dx} = e^{2x}\).
3. Multiply the entire differential equation by the integrating factor:
   • \(e^{2x} \left(\frac{dy}{dx} + 2y\right) = e^{2x} \sin 2x\)
   • This simplifies to \(\frac{d}{dx}(y e^{2x}) = e^{2x} \sin 2x\)
4. Integrate both sides with respect to \(x\):
   • \(y e^{2x} = \int e^{2x} \sin 2x \, dx\)
5. To solve the integral on the right, using integration by parts:
   • Let \(u = \sin 2x\) and \(dv = e^{2x} \, dx\)
   • This gives \(du = 2\cos 2x \, dx\) and \(v = \frac{1}{2}e^{2x}\)
   • So, \(\int e^{2x} \sin 2x \, dx = \frac{1}{2}e^{2x} \sin 2x - \int \frac{1}{2}e^{2x} \cdot 2\cos 2x \, dx\)
   • Continuing integration by parts again, we find:
   • \(=\frac{1}{2}e^{2x} \sin 2x-\frac{1}{4}e^{2x} \cos 2x + C\)
6. Hence, \(y e^{2x} = \frac{1}{2}e^{2x} \sin 2x - \frac{1}{4}e^{2x} \cos 2x + C\)
7. Substitute the initial condition \(y(0) = \frac{3}{4}\) to find \(C\):
   • \(y(0)e^{0} = \frac{1}{2}\sin 0 - \frac{1}{4} \cos 0 + C = \frac{3}{4}\)
   • This simplifies to \(0 - \frac{1}{4} + C = \frac{3}{4}\)
   • Solve for \(C\): \(C = 1\)
8. Now plug \(C = 1\) back into the solution:
   • \(y e^{2x} = \frac{1}{2}e^{2x} \sin 2x - \frac{1}{4}e^{2x} \cos 2x + 1\)
9. Calculate \(y\left(\frac{\pi}{8}\right)\):
   • \(y\left(\frac{\pi}{8}\right) = \left(\frac{1}{2} \sin \frac{\pi}{4} - \frac{1}{4} \cos \frac{\pi}{4} + \frac{1}{e^{\frac{\pi}{4}}}\right)\)
   • Substitute values: \(y\left(\frac{\pi}{8}\right) = e^{-\frac{\pi}{4}}\)

Thus, the value of \(y\left(\frac{\pi}{8}\right)\) is \(e^{-\frac{\pi}{4}}\).