Question

# Find the complete solution to the following differential equations. a) y''+2y'-y=10 b) 2y''- y = 3x^2

Find the complete solution to the following differential equations.

a) y''+2y'-y=10

b) 2y''- y = 3x^2

a) Given differential equation is : y''+2y'-y = 10

i.e., D2y+2Dy-y = 10

i.e., (D2+2D-1)y = 10

Let the trial solution be y = Aemx.

Then the auxiliary equation is m2+2m-1 = 0 which gives .

Therefore the general solution is : ,i.e., , where A, B are constants.

Next we have, for a particular integral,

i.e.,

i.e.,

i.e.,

i.e.,

i.e.,

i.e.,

i.e.,

Therefore, the complete solution is : , where A, B are constants.

b) Given differential equation is : 2y''-y = 3x2

i.e., 2D2y-y = 3x2

i.e., (2D2-1)y = 3x2

Let the trial solution be y = Aemx.

Then the auxiliary equation is 2m2-1 = 0 which gives .

Therefore the general solution is : , where A, B are constants.

Next we have, for a particular integral,

i.e.,

i.e.,

i.e.,

i.e.,

i.e.,

i.e.,

i.e.,

Therefore, the complete solution is : , where A, B are constants.

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