Solve the ODE :4 3 0y y y′′ ′− + =subject to the initial conditions y(0) = 1 andy'(0) = 2

To solve the given ordinary differential equation (ODE), we will follow these steps:

Step 1: Write down the ODE in standard form: 4y''' - 3y'' + 0y' - y = 0

Step 2: Determine the characteristic equation by assuming a solution of the form y = e^(rt), where r is a constant: 4r^3 - 3r^2 + 0r - 1 = 0

Step 3: Solve the characteristic equation to find the roots r1, r2, and r3. These roots will determine the form of the general solution.

Step 4: Once we have the roots, we can write the general solution of the ODE as: y(t) = C1e^(r1t) + C2e^(r2t) + C3e^(r3t)

Step 5: Apply the initial conditions y(0) = 1 and y'(0) = 2 to find the specific values of the constants C1, C2, and C3.

Step 6: Substitute the values of C1, C2, and C3 into the general solution to obtain the particular solution of the ODE.

Following these steps, we can solve the given ODE and find the solution that satisfies the initial conditions.