B solve for the general solution of the differential...

B. Solve for the general solution of the differential equations: (1) (D^2 - 5D + 4)y = 0 (2) (6D^2 - 5D - 6)y = 0 (3) (D^3 - 4D^2 + 4D)y = 0 (4) (D^4 - D^3)y = 0 (5) (4D^4 + 4D^3 - 3D^2 - 2D + 1)y = 0 C. Solve for the general solution of the differential equations: (1) (D^2 - 2D + 2)y = 0 (2) (D^3 + 2D^2 + D + 2)y = 0 (3) (D^3 + 7D^2 + 19D + 13)y = 0 (4) (D^3 + 100)^2y = 0 (5) (D^6 + 18D^4 + 81D^2)y = 0

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Transcript

00:01 Hello, let's have a look at the question.

00:03 So now for the first part we have to here find the general solutions.

00:07 So we have d square minus 5d plus 4 into y is equals to 0.

00:16 So the auxiliary equation will be equals to m square minus 5m plus 4 is equals to 0.

00:24 So from here we can write m square minus 4m minus m plus 4 is equals to 0.

00:33 Taking m common we get m minus 4 here minus 1 and m minus 4 is equals to 0.

00:42 So from here we get m minus 4 into m minus 1 is equals to 0.

00:48 That is m is equal to 1 or 4.

00:53 Therefore we can say that the general solution for this will be equals to yx is equals to c1, e to the power x plus c2, e to the power 4x.

01:10 Now for the second part, we have 6d square minus 5d minus 6 into y, minus 6 into y, is.

01:20 Equals to 0.

01:23 So now the auxiliary equation will be as 6m square minus 5m minus 6 is equals to 0.

01:32 So here we can write 6m square minus 9m plus 4m minus 6 is equals to 0.

01:42 So here let us take m common so we have 6m minus 9 plus 2 and here we have 2m 2m minus 3 is equals to 0.

01:55 Now here let us take 3 common also so we have 3m into 2m minus 3 plus 2m minus 3 is equals to 0.

02:08 So from here we get 2m minus 3 into 3m plus 2 is equals to 0...

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