Hi, can someone please help me to solve this?
Sorry to have to tell you, but this is way off the mark. You can always check the answer to a differntial equation problem by seeing whether it satisfies the given equation, and this one certainly does not. If then , and . If you plug these formulas for y" and y into the expression , you certainly don't get 0.
The way to solve an equation of this sort is to forget about the initial conditions y(0)=1 and y'(0)=1 entirely, until the very end of the solution.
Write the differential equation as , and look first at the so-called homogeneous equation that you get when you replace the -20 on the right-hand side by 0. The general solution to the equation is . You then have to look for a particular solution to the equation with the -20 on the right-hand side. There is an obvious solution in this case, namely to take y to be the constant function .
Add the solution to the homogeneous equation to the particular solution, and you get . This is the general solution to your equation. To finish the job, you have to choose the constants A and B so that the initial conditions y(0)=1 and y'(0)=1 are satisfied. (And if you want to finish the job properly, you should then differentiate your answer twice, and check that it really does satisfy the differential equation.)