Problem 20 - Advanced Engineering Mathematics

Question

In Problems 13-28, use the procedures developed in this chapter to find the general solution of each differential equation. $$ y^{\prime \prime}-2 y^{\prime}+y=x^{2} e^{x} $$

Step-by-Step Solution

Answer

The general solution is \( y = (C_1 + C_2 x)e^x + x^2 e^x \).

Step 1: Identify Type of Differential Equation

The given differential equation is \( y'' - 2y' + y = x^2 e^x \). This is a second-order, linear, non-homogeneous differential equation.

Step 2: Solve the Homogeneous Equation

First, solve the associated homogeneous equation: \( y'' - 2y' + y = 0 \). The characteristic equation is \( r^2 - 2r + 1 = 0 \), which has a double root \( r = 1 \). Thus, the general solution to the homogeneous equation is \( y_h = (C_1 + C_2 x)e^x \).

Step 3: Solve the Non-Homogeneous Equation

Since the non-homogeneous term is \( x^2 e^x \) and this contains a solution of the homogeneous equation, we use the method of undetermined coefficients. Assume a particular solution of the form \( y_p = Ax^2 e^x + Bxe^x + Ce^x \).

Step 4: Compute Derivatives of Particular Solution

Find the first and second derivatives of the particular solution \( y_p \): \( y_p' = (Ax^2 + 2Ax + B)e^x + (Bx + C)e^x \) and \( y_p'' = (Ax^2 + 4Ax + 2A + 2B)e^x + (Bx + C)e^x \).

Step 5: Substitute into Differential Equation

Substitute \( y_p, y_p', \) and \( y_p'' \) into the non-homogeneous differential equation to get:\((Ax^2 + 4Ax + 2A + 2B) + (Bx + C) - 2(Ax^2 + 2Ax + B) - 2(Bx + C) + (Ax^2 e^x + Bxe^x + Ce^x) = x^2 e^x.\)

Step 6: Simplify the Equation

By simplifying, we gather like terms and equate coefficients with the non-homogeneous part \(x^2 e^x\). We get the equations for coefficients: \( A = 1, \, B = 0, \, C = 0.\)

Step 7: Write General Solution

Combine the solutions to the homogeneous and particular equations: \( y = y_h + y_p = (C_1 + C_2 x)e^x + x^2 e^x.\)

Key Concepts

Second-Order Differential Equations

Second-order differential equations are a type of equation that involves the second derivative of a function. They are called "second-order" because the highest derivative present in the equation is the second derivative.
These equations are often used to model physical phenomena, such as motion under the influence of forces, electrical circuits, and more.
 

• An example of a second-order differential equation is \( y'' - 2y' + y = x^2 e^x \).
• This equation includes both the original function \( y \) and its first (\( y' \)) and second (\( y'' \)) derivatives.
• Second-order differential equations can be either homogeneous or non-homogeneous, and they can be linear or nonlinear.

Understanding these kinds of equations is essential in fields like engineering and physics, where describing how things change is crucial.

Non-Homogeneous Differential Equations

A non-homogeneous differential equation has terms that are not only dependent on the function and its derivatives but also include additional terms that are independent of the function.
For example, in our equation \( y'' - 2y' + y = x^2 e^x \), \( x^2 e^x \) is the non-homogeneous part.

• The presence of the non-homogeneous term means the equation cannot be solved simply by using solutions to its homogeneous counterpart.
• Instead, we must find a particular solution that directly addresses the non-homogeneous term in addition to solving the homogeneous equation.
• The general solution to a non-homogeneous differential equation is the sum of the solutions of its homogeneous equation and a particular solution to the non-homogeneous equation.

This approach allows us to account for additional influences or inputs represented by the non-homogeneous term.

Method of Undetermined Coefficients

The method of undetermined coefficients is used to find a particular solution to non-homogeneous linear differential equations. This technique is applicable when the non-homogeneous part is a specific type of function, such as polynomials, exponentials, sines, and cosines.
In our equation \( y'' - 2y' + y = x^2 e^x \), the method of undetermined coefficients helps find a particular solution to the non-homogeneous part \( x^2 e^x \).

• This method involves guessing a form for the particular solution, then determining the necessary coefficients to satisfy the equation.
• In this case, we assume a trial solution of the form \( y_p = Ax^2 e^x + Bxe^x + Ce^x \).
• We then substitute this guess back into the original equation, equate coefficients for like terms, and solve for \( A, B, \) and \( C \).

This systematic approach makes it possible to manage even complex differential equations by simplifying them into more workable parts.