Differential Equations- find the general solution(linear equation of order n)

Question

1. (D³ − 3D² − 10D) y = 0

2. (D³ − 3D² − 4D − 12) y = 0

3. d³x/dt³+d²x/dt²-2dx/dt = 0

4. d³x/dt²-19dx/dt+30 = 0

5. (4D³ − 21D − 10 ) y = 0

6. (D³ − 14D + 8) y= 0

7. (D³ − D² − 4D− 2) y = 0

8. (4D⁴ − 8D³ − 7D² + 11D + 6) y = 0

9. (4D⁴ − 16D³ + 7D² + 4D − 2) y = 0

10. (4D⁴ + 4D³ − 13D² − 7D + 6) y = 0

Answer 1

For this group of differential equations, the solution is to find the roots of polynomials. D is just an operator for dy/dx or dy/dt, depending on the problem. For each root, the component will be Ce^rt or Ce^rx. If there are duplicate roots, you need to multiply by a variable, for example C₁e^rt + C₂te^rt. If you get complex roots, you will probably need to convert to trig functions which should be explained in your textbook.

For example on number 1, the D-polynomial factors into D(D-5)(D+2)=0, giving r₁=0, r₂=5 and r₃=-2. Therefore,

y = C₁e^0t+C₂e^5t+C₃e^-2t Note: e^0t = 1

Problem 4 seems to have a typo.

The challenge here is going back to Algebra I to factor the polynomials.