00:01
So here in this question we have to find out the particular solution of the differential equations.
00:07
So here we are given the differential equation that is y double dash plus 4 of y is equal to x raised to the power 2 sine of 2 of x plus cos of 2 of x plus e raised to the power 2 of x plus 4.
00:21
So from here, its characteristic equation from here will be equals to r to the power 2 plus 4.
00:31
That is equals to 0 from here.
00:34
So we can say that the roots of this characteristic equation is equal to r, that is plus minus 2 of iota.
00:40
Therefore, it's homogenous equation, solution of the equation, y is equals to c1, cause of 2 of x, plus c2, sine of 2 of x.
00:52
So to find out its particular solution, we can say that we need to consider the right hand side of the equation so we have sign of 2 of x and cause of 2 of x so this both are the solution of homogeneous equations so we can say that therefore we need to use the matter of undetermined coefficient with the modification so here we can say that y from here is x multiplied by the a sign of 2 of x plus b cause of 2 x so we have also e raised to the power 4 of x which is the solution of the homogeneous.
01:32
So, y from here is equal to c, e raised to the power 4 of x...