QUESTIONS 1) Find the general solution of the following differential equation (30 points) y''+25y=cos(5x) 2) Solve the initial value problem by using Laplace transform (30 points) (d^2y)/(dt^2)+6(dy)/(dt)-7y=0 y(0)=1, y'(0)=-2 3) Find the power series solutions in powers of x in the following differential equation (40 points) (d^2y)/(dx^2)+3x(dy)/(dx)-4y=0

1) To find the general solution of the differential equation , we first find the complementary solution by solving the homogeneous equation . The characteristic equation is , which has complex roots . Therefore, the complementary solution is .Next, we find a particular solution of the form . Substituting this into the original equation, we get . Simplifying, we get , which implies . Therefore, the particular solution is .The general solution is .2) To solve the initial value problem using Laplace transform, we take the Laplace transform of both sides of the differential equation . Applying the Laplace transform to each term, we get , where is the Laplace transform of .Substituting the initial conditions and , we get . Simplifying, we get . Solving for , we get .To find , we take the inverse Laplace transform of . Using partial fraction decomposition, we get . Therefore, .3) To find the power series solutions of the differential equation , we assume a power series solution of the form . Substituting this into the differential equation, we get .To simplify the equation, we shift the indices of the first two sums by 2 and 1, respectively. This gives us $\sum_{n=2}^{\infty}(n-2)(n-1)a_{n-2}x^{n-2}+3\sum_{n=1}^{\infty}na_{n-1}x^{n}