Problem 7 - Advanced Engineering Mathematics

Question

In Problems 1-26, solve the given differential equation by undetermined coefficients. $$ y^{\prime \prime}+3 y=-48 x^{2} e^{3 x} $$

Step-by-Step Solution

Answer

The general solution is a combination of the complementary and particular solutions.

Step 1: Identify the Homogeneous Equation

First, we identify the complementary equation associated with the given differential equation, which is obtained by setting the right-hand side to zero: \[ y'' + 3y = 0 \]

Step 2: Solve the Homogeneous Equation

We solve the characteristic equation for the homogeneous part: \[ r^2 + 3 = 0 \]Solving gives roots as \[ r = \pm \sqrt{3}i \]. Hence, the complementary solution is \[ y_c(x) = c_1 \cos(\sqrt{3}x) + c_2 \sin(\sqrt{3}x) \].

Step 3: Formulate the Particular Solution

Determine the form of the particular solution for the non-homogeneous term \(-48x^2e^{3x}\). Since the non-homogeneous term contains \(e^{3x}\), try using \(y_p(x) = (Ax^2 + Bx + C) e^{3x}\).

Step 4: Calculate Derivatives of the Particular Solution

Find \(y'_p\) and \(y''_p\) to substitute into the differential equation:\[ y'_p(x) = (2Ax + B)e^{3x} + 3(Ax^2 + Bx + C)e^{3x} \]\[ y''_p(x) = 2Ae^{3x} + 6Ax e^{3x} + 3(2Ax + B)e^{3x} + 9(Ax^2 + Bx + C)e^{3x} \]

Step 5: Substitute into Original Equation

Substitute \(y_p\), \(y'_p\), and \(y''_p\) into the original differential equation,\[ y''_p + 3y_p = -48x^2e^{3x} \].After simplification, this gives \[ (12A + 3B)x^0e^{3x} + (15A + 6B)x^1e^{3x} + (9A - 48)x^2e^{3x} = 0 \].

Step 6: Solve for Coefficients

Set coefficients of \(x^0\), \(x^1\), and \(x^2\) separately equal to zero to solve for A, B, and C:1. \(9A - 48 = 0\), solve for \(A\).2. \(15A + 6B = 0\), solve for \(B\).3. \(12A + 3B + 3C = 0\), solve for \(C\).

Step 7: Write the General Solution

With the constants A, B, and C found, the particular solution is determined. The general solution to the differential equation is \[ y(x) = y_c(x) + y_p(x) \]\[ = c_1 \cos(\sqrt{3}x) + c_2 \sin(\sqrt{3}x) + (Ax^2 + Bx + C)e^{3x} \].

Key Concepts

undetermined coefficients

Undetermined coefficients is a method used to find particular solutions of non-homogeneous linear differential equations. It provides a systematic approach to guessing the form of a particular solution based on the function on the right-hand side of the equation.

To apply this method, we first identify the complementary solution of the associated homogeneous equation, and then guess a form for the particular solution that will account for the non-homogeneous term. This guess is often based on the type of function we have on the right side, such as a polynomial, exponential, sine, or cosine function. We then determine the unknown coefficients by plugging the guessed form into the differential equation and matching coefficients.

• Key insight: For polynomials, use a polynomial of the same degree.
• For exponentials, use a similar exponential form.
• Sines and cosines often require both functions in the guessed solution.

This method simplifies solving non-homogeneous differential equations when the non-homogeneous part matches certain common function types.

complementary solution

The complementary solution, often denoted as \( y_c(x) \), is the solution to the associated homogeneous differential equation. This is the part of the solution that addresses the natural behavior of the system without external forces or inputs.

When we set the right-hand side of the differential equation to zero, we get the homogeneous equation. For example, if we have a differential equation like \( y'' + 3y = 0 \), the complementary solution addresses this simplified form.

To find it, we solve the characteristic equation derived from the homogeneous part. If we have a second-order linear differential equation, the characteristic equation is often of the form \( ar^2 + br + c = 0 \). Solving this gives us the roots, which in turn allow us to express the complementary solution.

• If roots are real and distinct, the solution is an exponential form.
• For complex roots, trigonometric functions like sine and cosine are used.
• Repeated roots lead to solutions involving powers of \( x \).

Understanding the complementary solution is key to capturing the intrinsic behavior of the differential equation.

particular solution

The particular solution, written as \( y_p(x) \), complements the complementary solution to provide a complete general solution for a non-homogeneous differential equation. It specifically accounts for the non-homogeneous part of the equation, typically the function that represents an external input.

In the process of finding \( y_p(x) \), we effectively engage in a trial-and-error method, called the method of undetermined coefficients, to guess a functional form that would satisfy the differential equation. This form should not be part of the complementary solution, ensuring that it uniquely responds to the non-homogeneous function.

Once a form is guessed, it involves taking its derivatives and substituting these back into the original equation. The result is a system of equations that is solved for the coefficients of the guessed function.

• Ensures the particular solution accurately offsets the non-homogeneous part.
• The choice of solution form is crucial for practical success.
• Some trial and error might be necessary to pinpoint the precise form.

Thus, the particular solution allows the differential equation to be fully satisfied when combined with the complementary solution.

non-homogeneous equation

A non-homogeneous differential equation is one which includes a non-zero function on the right-hand side, often representing an external force or influence acting on the system. These are equations of the form \( y = f(x) \) plus some function \( g(x) \), where \( g(x) eq 0 \).

The presence of \( g(x) \) distinguishes non-homogeneous equations from their homogeneous counterparts. It fully determines why such equations require both a complementary solution and a particular solution.

The combined result from these solutions yields the general solution, which effectively captures both intrinsic system dynamics and external influences.

• Non-homogeneous equations may reflect real-world scenarios such as mechanical systems subjected to external forces.
• They necessitate additional techniques to manage the non-zero \( g(x) \).
• Understanding the nature of \( g(x) \) is crucial to selecting the correct method for solving the equation.

Grasping the nature of a non-homogeneous equation is essential for successfully resolving complex differential scenarios.