Laplace Transform: Initial Value Problems (IVP)

A. Use Laplace transform to solve the following initial value problems:

  1. \(y''-6y'+5y=0; \quad y(0)=2, \: y'(0)=5\) View Solution
  2. \(y^{(4)}-y=0; \quad y(0)=1,\: y'(0)=0,\: y''(0)=1, y'''(0)=0\) View Solution
  3. \( y^{(4)}-4y'''+6y''-4y'+y=0, \quad y(0)=0, \: y'(0)=1, \: y''(0)=0, \: y'''(0)=1\) View Solution
  4. \(y''-2y'+2y=e^{-t}; \quad y(0)=0, \: y'(0)=1\) View Solution
  5. \(y'+5y=f(t); \quad y(0)=0, \quad f(t)=\begin{cases} 1, & 0 \leq t < 1 \\ 0, & t \geq 1 \end{cases} \) View Solution
  6. \(y''+4y=3\sin t; \quad y(0)=1,\: y'(0)=-1 \) View Solution
  7. \(y''+2y'-15y=6\delta(t-1); \quad y(0)=0,\: y'(0)=2 \) View Solution
  8. \(y''+y=\delta(t-\pi) +1; \quad y(0)=0, \: y'(0)=0\) View Solution
  9. \( y''-2y'-3y=2\delta(t-1)-\delta(t-3);\quad y(0)=2, \: y'(0)=2\) View Solution
  10. \(y''+4y=\sin (t) -u(t-2\pi)\sin(t-2\pi), \quad y(0)=0, \: y'(0)=0 \) View Solution
  11. \(y''+4y=g(t); \quad y(0)=0, \: y'(0)=0, \quad g(t)=\begin{cases} 0, & 0\leq t < 5 \\ \frac{1}{5} (t-5), & 5 \leq t < 10 \\ 1, & t \geq 10 \end{cases} \) View Solution
  12. \(y''+4y = \cos (\alpha t); \quad y(0)=0, \: y'(0)=1\) View Solution

B. Volterra Integral Equation

Solve by the Laplace transform: \(\displaystyle{y(t)-\int_0^t (t-v)y(v)\, dv = 1.} \) View Solution

C. Integro-differential Equation

Solve by the Laplace transform: \(\displaystyle{y'(t)-\frac{1}{2}\int_0^t (t-v)^2y(v)\, dv = -t, \quad y(0)=1.}\) View Solution