Solve the differential equation using variation of parameters: $y^{\text{'}\text{'}}-2y^{\text{'}}+y=e^t\arctan \left(t\right)$. Which of the following is the general solution?

• A. $y\left(t\right)=(C_1+C_{2}t)e^t+e^t\int (t-s)e^{-s}\arctan \left(s\right)ds$
• B. $y\left(t\right)=(C_1+C_{2}t)e^t+e^t\int e^{-t}\arctan \left(t\right)dt$
• C. $y\left(t\right)=C_1{e}^t+C_{2}t{e}^t+e^t\arctan \left(t\right)$
• D. $y\left(t\right)=(C_1+C_{2}t)e^t+e^t\int \arctan \left(t\right)dt$

Answer: (A)

Verified step by step guidance

1. Step 1: Identify the type of differential equation. The given equation y'' - 2y' + y = e^t arctan(t) is a second-order linear non-homogeneous differential equation. The method of variation of parameters is suitable for solving such equations.
2. Step 2: Solve the corresponding homogeneous equation. The homogeneous equation is y'' - 2y' + y = 0. Solve this by assuming a solution of the form y_h = e^{rt}, where r is a constant. Substitute y_h into the homogeneous equation to find the characteristic equation: r^2 - 2r + 1 = 0. Solve this quadratic equation to find the roots.
3. Step 3: Write the general solution of the homogeneous equation. Since the characteristic equation has a repeated root (r = 1), the general solution of the homogeneous equation is y_h = (C_1 + C_2 t)e^t, where C_1 and C_2 are constants.
4. Step 4: Apply the method of variation of parameters. For the non-homogeneous equation, assume a particular solution of the form y_p = u_1(t)y_1 + u_2(t)y_2, where y_1 = e^t and y_2 = te^t are the solutions of the homogeneous equation. The functions u_1(t) and u_2(t) are determined by solving the system of equations: u_1'(t)y_1 + u_2'(t)y_2 = 0 and u_1'(t)y_1' + u_2'(t)y_2' = e^t arctan(t).
5. Step 5: Solve for u_1(t) and u_2(t). Use the Wronskian W = y_1y_2' - y_1'y_2 to simplify the equations for u_1'(t) and u_2'(t). Integrate u_1'(t) and u_2'(t) to find u_1(t) and u_2(t). Substitute these into y_p = u_1(t)y_1 + u_2(t)y_2 to find the particular solution. Combine y_h and y_p to write the general solution: y(t) = (C_1 + C_2 t)e^t + e^t ∫(t-s)e^{-s}arctan(s)ds.