Solve d^2y/dx^2 - dy/dx - 2y = 2logx + 1/x + 1/x^2

Steps to Solve

1. Identify the Differential Equation First, write down the given second-order linear differential equation in standard form, typically written as: $$ a y'' + b y' + c y = f(t) $$ where ( f(t) ) is the non-homogeneous part.

2. Find the Complementary Solution (Homogeneous Solution) To find the complementary solution, solve the associated homogeneous equation: $$ a y'' + b y' + c y = 0 $$ This is generally done by finding the characteristic equation: $$ a r^2 + b r + c = 0 $$ Solve for the roots ( r ), which can be real or complex.

3. Construct the Complementary Solution Based on the roots of the characteristic equation:

• If the roots are real and distinct: $$ y_c = C_1 e^{r_1 t} + C_2 e^{r_2 t} $$
• If the roots are real and repeated: $$ y_c = (C_1 + C_2 t) e^{r t} $$
• If the roots are complex: $$ y_c = e^{\alpha t} (C_1 \cos(\beta t) + C_2 \sin(\beta t)) $$ where ( r_1, r_2 ) are the real roots and ( \alpha, \beta ) are from the complex roots ( \alpha + i \beta ).

4. Find a Particular Solution Next, to find the particular solution ( y_p ), you can use methods such as:

• Method of Undetermined Coefficients: Guess a specific form of ( y_p ) based on ( f(t) ).
• Variation of Parameters: Use the complementary solution to derive ( y_p ).

5. Combine Solutions The general solution of the differential equation is: $$ y(t) = y_c + y_p $$ Where ( y_c ) is the complementary solution and ( y_p ) is the particular solution.

6. Solve Initial or Boundary Conditions if Provided If there are specific initial or boundary conditions given, substitute them into the general solution to solve for any constants ( C_1, C_2, ) etc.