(D^2 + 4)y=sinx ; y(0)=2;y'(0)=-1 find general solution

The given differential equation is a second order non-homogeneous differential equation. The general solution of such an equation is the sum of the general solution of the homogeneous equation and a particular solution of the non-homogeneous equation.

Step 1: Solve the homogeneous equation (D^2 + 4)y = 0 The characteristic equation of the homogeneous equation is r^2 + 4 = 0. Solving this gives r = ±2i. Therefore, the general solution of the homogeneous equation is y_h = C1cos(2x) + C2sin(2x).

Step 2: Find a particular solution of the non-homogeneous equation We can guess a particular solution of the form y_p = Acos(x) + Bsin(x). Differentiating this twice and substituting into the non-homogeneous equation gives us a system of equations for A and B. Solving this system gives A = 0 and B = -1/2. Therefore, a particular solution is y_p = -1/2*sin(x).

Step 3: The general solution of the non-homogeneous equation is the sum of the general solution of the homogeneous equation and the particular solution of the non-homogeneous equation. Therefore, the general solution is y = C1cos(2x) + C2sin(2x) - 1/2*sin(x).

Step 4: Use the initial conditions to find the constants C1 and C2 We are given that y(0) = 2 and y'(0) = -1. Substituting these into the general solution and its derivative gives a system of equations for C1 and C2. Solving this system gives C1 = 2 and C2 = -1/2.

Therefore, the general solution of the given differential equation is y = 2cos(2x) - 1/2sin(2x) - 1/2*sin(x).

The given differential equation is a second order non-homogeneous differential equation. The general solution of such an equation is the sum of the general solution of the homogeneous equation and a particular solution of the non-homogeneous equation.

Step 1: Solve the homogeneous equation (D^2 + 4)y = 0 The characteristic equation of the homogeneous equation is r^2 + 4 = 0. Solving this gives r = ±2i. Therefore, the general solution of the homogeneous equation is y_h = C1cos(2x) + C2sin(2x).

Step 2: Find a particular solution of the non-homogeneous equation We can guess a particular solution of the form y_p = Acos(x) + Bsin(x). Differentiating this twice and substituting into the non-homogeneous equation gives us a system of equations for A and B. Solving this system gives A = 0 and B = -1/2. Therefore, a particular solution is y_p = -1/2*sin(x).

Step 3: The general solution of the non-homogeneous equation is the sum of the general solution of the homogeneous equation and the particular solution of the non-homogeneous equation. Therefore, the general solution is y = C1cos(2x) + C2sin(2x) - 1/2*sin(x).

Step 4: Use the initial conditions to find the constants C1 and C2 We are given that y(0) = 2 and y'(0) = -1. Substituting these into the general solution and its derivative gives a system of equations for C1 and C2. Solving this system gives C1 = 2 and C2 = -1.

Therefore, the general solution of the given differential equation is y = 2cos(2x) - sin(2x) - 1/2sin(x).