Solving the Differential Equation D^2 - 4y x^2cosh2x: A Comprehensive Guide

The differential equation D^2 - 4y x^2cosh2x can be solved using various methods, such as the method of undetermined coefficients or variation of parameters. This article provides a detailed guide to solving this particular equation step-by-step. We will cover the process from setting up the homogeneous equation to finding the particular solution and finally, constructing the general solution.

1. Solving the Homogeneous Equation

First, we tackle the homogeneous part of the equation:

D^2 - 4y_h 0

This can be factored as:

D - 2D - 2y_h 0

The characteristic equation is:

r^2 - 4 0 which simplifies to r - 2r - 2 0

The roots are r 2 and r -2. Thus, the general solution to the homogeneous equation is:

y_h C_1 e^{2x} C_2 e^{-2x}

Where C_1 and C_2 are constants.

2. Solving the Non-Homogeneous Equation

To find a particular solution y_p to the non-homogeneous equation:

D^2 - 4y x^2cosh2x

We consider the right-hand side x^2cosh2x. Since cosh2x can be expressed as ,

We can write:

y_p Ax^2 e^{2x} Bx^2 e^{-2x} Cx e^{2x} Dx e^{-2x} Ex^2 F

Where A, B, C, D, E, F are constants to be determined.

3. Computing D2y_p

We need to compute D2y_p and substitute it back into the original equation. This process involves differentiating y_p twice and then substituting into the left-hand side of the equation.

First Derivative:

y_p differentiate each term

Second Derivative:

y_p differentiate each term again

4. Substituting into the Equation

Substituting y_p and its second derivative into the left-hand side D2 - 4y_p:

D2 - 4y_p y_p - 4y_p

This should equal x^2cosh2x.

5. Collecting Like Terms and Solving for Constants

After substituting y_p and simplifying, you will have a polynomial in x and exponential functions. Match coefficients on both sides of the equation to solve for A, B, C, D, E, F.

6. Constructing the General Solution

Once you have y_p, the general solution to the original differential equation is:

y y_h y_p C_1 e^{2x} C_2 e^{-2x} y_p

Conclusion

The final solution will consist of the homogeneous solution plus the particular solution y_p found through the method of undetermined coefficients. The exact form of y_p will depend on the coefficients determined in the previous steps.

If you need further assistance with any specific calculations or steps, feel free to ask.