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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