Find the general solution for differential equation (D4 - 5D3 + 5D2 + 5D - 6)y = 0
1 answer:
Given differential equation, (D<span>4 </span>- 5D3<span> + 5D</span>2<span> + 5D - 6)y = 0</span>
=> For general solution of equation,
Solve D<span>4 </span>- 5D3<span> + 5D</span>2<span> + 5D - 6 = 0</span>
=> D<span>4 </span>- 5D3<span> + 6D</span>2<span> - D</span><span>2 </span>+ 5D - 6 = 0
=> D2<span> (D</span>2<span> - 5D + 6) - (D</span>2<span> - 5D + 6) = 0</span>
=> (D2<span> - 5D + 6)(D</span>2<span> - 1) = 0 ................................(1)</span>
<span>Now </span>
D<span>2 </span><span>- 1 = (D - 1)(D + 1) and </span>
Factors of D2<span> - 5D + 6</span>
D2<span> - 5D + 6 = D</span>2<span> - 2D - 3D + 6</span>
= D(D - 2) - 3(D - 2)
= (D - 3)(D - 2)
Therefore, equation (1) implies
(D2<span> - 5D + 6)(D</span>2<span> - 1) = (D - 3)(D - 2)(D - 1)(D + 1) = 0</span>
=> D = 3, 2, 1, -1 or D = -1, 1,, 2, 3
=> General solution of differential equation is,<span>
=><span> y = C1 e-x + C2 ex + C3 e2x + C4 e3x</span> .
</span>
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