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

  1.   \(y''-6y'+5y=0; \:\: y(0)=2, \: y'(0)=5\)    solution 


  2.   \(y^{(4)}-y=0; \:\: y(0)=1,\: y'(0)=0,\: y''(0)=1, y'''(0)=0\)    solution


  3.    \( y^{(4)}-4y'''+6y''-4y'+y=0, \quad y(0)=0, \: y'(0)=1, \: y''(0)=0, \: y'''(0)=1\)   solution


  4.    \(y''-2y'+2y=e^{-t}; \:\: y(0)=0, \: y'(0)=1\)    solution


  5.    \(y'+5y=f(t); \quad  y(0)=0,  \:  f(t)=\begin{cases} 1, \:  0 \leq t <1, \\ 0,  \:  t \geq 1  \end{cases} \)   solution


  6.  \(y''+4y=3\sin t; \quad y(0)=1,\: y'(0)=-1 \)   solution


  7.   \(y''+2y'-15y=6\delta(t-1); \quad y(0)=0,\: y'(0)=2 \)   solution


  8.   \(y''+y=\delta(t-\pi) +1; \quad y(0)=0, \: y'(0)=0\)   solution


  9.  \( y''-2y'-3y=2\delta(t-1)-\delta(t-3);\quad y(0)=2, \: y'(0)=2\)   solution 


  10.   \(y''+4y=\sin (t) -u(t-2\pi)\sin(t-2\pi), \quad y(0)=0, \: y'(0)=0 \)   solution


  11.   \(y''+4y=g(t); \quad  y(0)=0, \: y'(0)=0, \:    g(t)=\begin{cases} 0, \:  0\leq t <5,\\ \frac{1}{5} (t-5), \: 5 \leq t < 10, \\ 1, \:  t \geq 10 \end{cases} \)   solution


  12.   \(y''+4y = \cos (\alpha t); \quad  y(0)=0, \: y'(0)=1\)    solution 

 

B. Solve the Volterra integral equation by the Laplace transform: \(\displaystyle{y(t)-\int_0^t (t-v)y(v)\, dv = 1.} \)   solution

 

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