Higher order linear diff equations:

1. Find the general solution of the diff equations.   \(D=\dfrac{d}{dt} \)  solution

 a. \( y'''+y''-4y'-4y'=0 \)    

 b. \( y^{(4)}-5y''+4y=0 \)

 c. \( (D^2+D-6)(D^2+5)y=0 \)  

2. Find the solution of the given initial value problem.

  a. \( y'''+4y'=0,    y(0)=0, y'(0)=1, y''(0)=2\)  solution

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

3. Show that the general solution of \(y^{(4)}-y=0\) can be written as \(\displaystyle{ y=A\cos t + B\sin t+C\cosh t + D\sinh t}\).   solution

4. Given the list of roots and their multiplicities of the characteristic equation, find a general solution. What is the order of the corresponding ODE? (\( k_i = \) multiplicity of root \(r_i\))  solution

\[  r_1=-2, k_1=3; r_2=1, k_2=2; r_{3,4}=-1\pm 5i, k_{3,4}=1 \]

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