Question

If the solution of \[ \frac{d^2y}{dt^2} = -\frac54y+\frac{dy}{dt}, \] satisfying \[ y(0)=1, \qquad \left(\frac{dy}{dt}\right)_{t=0}=0, \] is \(y(t)\), then \(y(\pi)=\)

• A. \(e^{-2\pi}\)
• B. \(e^{\frac{\pi}{2}}\)
• C. \(5e^{-2\pi}\)
• D. \(4e^{\frac{\pi}{2}}\)

Hint:

For complex roots \(a\pm bi\), the solution is \[ y=e^{at}(C_1\cos bt+C_2\sin bt). \] Use initial conditions immediately to determine the constants.

Answer 1

The Correct Option is B