Homework #5

1. Solve the following equations (find the general solution):
   
   
   1. \( y'''-3y'-2y=te^{-t}\sin(3t)-4e^{-t}+5t^2-2 \) using the method of undetermined coefficients. Then find the solution whose initial values are \( y(0)=1, y'(0)=2, y''(0)=3 \).
      
      
   2. \( t^2y''+ty'+4y = \sin(\ln(t^2))+3\sqrt{t} \) using both the method of undetermined coefficients and variation of parameters.
      
      
   3. \( y''-4y'+3y=t^3+3 \) using the method of undetermined coefficients, variation of parameters, and by converting to an equivalent linear system.
      
      
   4. \( y'''-5y''+8y'-4y=e^t \) using the method of undetermined coefficients, variation of parameters, and by converting to an equivalent linear system.
      
      
2. Solve the following equations (find the general solution) -- for convenience assume \( t>0 \):
   
   
   1. \( t^2y''+2ty'-6y = e^{2t} \) then find the solution whose initial values are \( y(1)=2, y'(1)=3 \).
      
      
   2. \( y'''-3y''+2y' = 2t^{-3}+3t^{-2}+2t^{-1} \)
      
      
   3. \( y'''-y''+9y'-9y = g(t) \) where \( g(t) \) is a continous function.
      
      
      
3. Find the general solution of the system: \( \begin{array}{ccccc} x_1' & = & -x_1 & & \\ x_2' & = & -x_1 & & +3x_3 \\ x_3' & = & 3x_1 & -3x_2 & \end{array} \). Then solve the corresponding initial value problem where \( x_1(0)=0, x_2(0)=1, x_3(0)=1 \).
   
   
   
4. Find the general solution of the system: \( \begin{array}{cccccc} x_1' & = & x_1 & +x_2 & +x_3 & +1 \\ x_2' & = & & -x_2 & & & \\ x_3' & = & -2x_1 & -x_2 & -2x_3 & +2e^{-t} \end{array} \). Then solve the corresponding initial value problem where \( x_1(0)=3, x_2(0)=2, x_3(0)=1 \).
   
   
   
5. Find the general solution of the system: \( \begin{array}{cccccc} x_1' & = & x_1 & -x_2 & +3x_3 & -2t \\ x_2' & = & -x_1 & +2x_2 & +2x_3 & \\ x_3' & = & -x_1 & & +4x_3 & +2t \end{array} \). Then solve the corresponding initial value problem where \( x_1(0)=0, x_2(0)=1, x_3(0)=0 \).