To solve a differential equation in the form a d²y/dx² + b dy/dx + cy = 0, you can use various methods depending on the values of a, b, and c. Here is a step-by-step approach:
1. Identify the type of differential equation:
- If a, b, and c are constants and a ≠ 0, b ≠ 0, c ≠ 0, then it is a homogeneous second-order linear differential equation.
- If a, b, and c are functions of x, then it is a non-homogeneous second-order linear differential equation.
2. Solve the homogeneous equation:
- Assume a solution of the form y = e^(mx), where m is a constant.
- Substitute this solution into the differential equation and simplify.
- The resulting characteristic equation is am² + bm + c = 0.
- Solve the characteristic equation to find the values of m.
- Depending on the roots of the characteristic equation, the general solution for the homogeneous equation is:
- If the roots are real and distinct: y = c₁e^(m₁x) + c₂e^(m₂x)
- If the roots are real and equal: y = (c₁ + c₂x)e^(mx)
- If the roots are complex: y = e^(ax)(c₁cos(bx) + c₂sin(bx))
3. Solve the non-homogeneous equation (if applicable):
- Determine the form of the particular solution based on the non-homogeneity.
- If f(x) is a polynomial of degree n, then the particular solution is a polynomial of the same degree.
- If f(x) is a trigonometric function, then the particular solution is a sum of trigonometric functions.
- If f(x) is an exponential function, then the particular solution is an exponential function.
- Substitute the particular solution into the differential equation and solve for the coefficients.
- The general solution for the non-homogeneous equation is the sum of the homogeneous solution and the particular solution.
4. Apply any initial conditions:
- If initial conditions are given (e.g., y(0) = 1, y'(0) = 2), substitute them into the general solution.
- Solve for the constants using the given initial conditions.
5. Simplify and write the final solution:
- Plug the values of the constants obtained into the general solution.
- Simplify the expression if possible.
- Write the final solution in a clear and concise form.
Remember to always double-check your solution and make sure it satisfies the original differential equation. Differential equations can have multiple solutions, so it's important to verify your answer.
I hope this explanation helps you understand how to solve a differential equation in the given form. Let me know if you have any further questions!