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

  1.   y    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