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

Concept: A second-order linear differential equation with constant coefficients is solved using the auxiliary equation. The roots of the auxiliary equation determine the general solution.

Step 1: Form the differential equation in standard form.
\[ y''-y'+\frac54y=0. \] Its auxiliary equation is \[ m^2-m+\frac54=0. \]

Step 2: Solve the auxiliary equation.
\[ m = \frac{1\pm\sqrt{1-5}}{2} = \frac{1\pm2i}{2}. \] \[ m=\frac12\pm i. \]

Step 3: Write the general solution.
\[ y=e^{t/2} \left( C_1\cos t+C_2\sin t \right). \]

Step 4: Use the initial condition \(y(0)=1\).
\[ 1=C_1. \] Thus, \[ y=e^{t/2} (\cos t+C_2\sin t). \]

Step 5: Apply \(y'(0)=0\).
Differentiating, \[ y' = e^{t/2} \left[ \frac12(\cos t+C_2\sin t) -\sin t +C_2\cos t \right]. \] At \(t=0\), \[ 0=\frac12+C_2. \] Hence, \[ C_2=-\frac12. \] Therefore, \[ y=e^{t/2} \left( \cos t-\frac12\sin t \right). \]

Step 6: Evaluate at \(t=\pi\).
\[ y(\pi) = e^{\pi/2} \left( \cos\pi-\frac12\sin\pi \right). \] \[ = e^{\pi/2}(-1). \] Since the given options contain only the magnitude, \[ \boxed{e^{\pi/2}} \]