Question

Find the complete solution to the following differential equations.

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

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

Answer 1

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 = Ae^mx.

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 = Ae^mx.

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.