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Solve the given differential equation by undetermined coefficients. y" - 12y' + 36y = 12x + 6 y(x)

Solve the given differential equation by undetermined coefficients. y" - 12y' + 36y = 12x + 6 y(x)

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.1: Solutions Of Elementary And Separable Differential Equations
Problem 9E: Find the general solution for each differential equation. Verify that each solution satisfies the...
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## Solving Differential Equations by Undetermined Coefficients ### Problem Statement Solve the given differential equation by the method of undetermined coefficients: \[ y'' - 12y' + 36y = 12x + 6 \] \[ y(x) = \_\_\_\_\_\_\_\_\_ \] ### Explanation In this problem, we are tasked with solving a non-homogeneous second-order linear differential equation using the method of undetermined coefficients. This method is particularly useful for solving linear differential equations with constant coefficients and a specific type of non-homogeneous term. Steps involved in solving the differential equation: 1. **Solve the Homogeneous Equation**: First, find the general solution to the associated homogeneous differential equation: \[ y'' - 12y' + 36y = 0 \] 2. **Find the Particular Solution**: Next, we find a particular solution to the original non-homogeneous differential equation. The form of the particular solution depends on the non-homogeneous term \(12x + 6\). 3. **Combine Both Solutions**: The general solution of the original non-homogeneous differential equation is the sum of the general solution of the homogeneous equation and the particular solution. ### Detailed Steps 1. **Homogeneous Solution**: The characteristic equation for \( y'' - 12y' + 36y = 0 \) is: \[ r^2 - 12r + 36 = 0 \] Solving for \(r\), we get: \[ (r - 6)^2 = 0 \] So, \( r = 6 \) (a repeated root). Therefore, the general solution to the homogeneous equation is: \[ y_h(x) = (C_1 + C_2x)e^{6x} \] 2. **Particular Solution**: Assume a particular solution of the form \( y_p(x) = Ax + B \) since the non-homogeneous term is \(12x + 6\). Differentiate \( y_p \): \[ y_p'(x) = A \] \[ y_p''(x) = 0 \] Substitute \( y_p(x) \), \( y_p'(x) \), and \( y_p''(x) \) into the original
Transcribed Image Text:## Solving Differential Equations by Undetermined Coefficients ### Problem Statement Solve the given differential equation by the method of undetermined coefficients: \[ y'' - 12y' + 36y = 12x + 6 \] \[ y(x) = \_\_\_\_\_\_\_\_\_ \] ### Explanation In this problem, we are tasked with solving a non-homogeneous second-order linear differential equation using the method of undetermined coefficients. This method is particularly useful for solving linear differential equations with constant coefficients and a specific type of non-homogeneous term. Steps involved in solving the differential equation: 1. **Solve the Homogeneous Equation**: First, find the general solution to the associated homogeneous differential equation: \[ y'' - 12y' + 36y = 0 \] 2. **Find the Particular Solution**: Next, we find a particular solution to the original non-homogeneous differential equation. The form of the particular solution depends on the non-homogeneous term \(12x + 6\). 3. **Combine Both Solutions**: The general solution of the original non-homogeneous differential equation is the sum of the general solution of the homogeneous equation and the particular solution. ### Detailed Steps 1. **Homogeneous Solution**: The characteristic equation for \( y'' - 12y' + 36y = 0 \) is: \[ r^2 - 12r + 36 = 0 \] Solving for \(r\), we get: \[ (r - 6)^2 = 0 \] So, \( r = 6 \) (a repeated root). Therefore, the general solution to the homogeneous equation is: \[ y_h(x) = (C_1 + C_2x)e^{6x} \] 2. **Particular Solution**: Assume a particular solution of the form \( y_p(x) = Ax + B \) since the non-homogeneous term is \(12x + 6\). Differentiate \( y_p \): \[ y_p'(x) = A \] \[ y_p''(x) = 0 \] Substitute \( y_p(x) \), \( y_p'(x) \), and \( y_p''(x) \) into the original
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