therefore:can anyone give msom light on this question, thanks

find the complete solution of the differential equation

d2y/dx2 – 4 dy/dx + 3y = 65 sin 2 x

Mr F says: It's expected that you can find the homogeneous solution, that is, the solution to .

So I'll assume that your trouble is in getting a particular solution.

It's unclear from your notation whether you mean or .

If then try as a particular solution.

If then note that and so you should try as a particular solution.

In each case you substitute the particular solution into the DE and solve for the values of a, b, c .......

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find the value of the constantasuch thaty= axe^-1is a solution of the differential equation Mr F says: You meany= axe^-x.

d2y/dx2 + 3 dy/dx + 2y = 2 e-x

Mr F says: You're being asked this question because it's the form of a particular solution to the DE.

The general solution to the DE is the homogeneous solution plus the particular solution.

It's expected that you can find the homogeneous solution. It's .

Then the general solution is . The value of a is found below.

find the solution for which y=1 and dy/dx =3 when x=0

Mr F says: Substitute these boundary conditions into the general solution to get the value of A and B.

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Substitute these expressions into the DE and simplify:

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