tHe hon-homogeneous problem y" + y' - 2y = 36e-2z First we consider the homogeneous problem y" + y' - 2y = 0: 1) the auxiliary equation is ar?+ br + c= ^2+r-2 = 0. 2) The roots of the auxiliary equation are 1, -2 (enter answers as a comma separated list). 3) A fundamental set of solutions is e^x, e^(-2x) (enter answers as a comma separated list). Using these we obtain the the complementary solution y. = c¡Y1 + c2Y2 for arbitrary constants Cq and c2- Next we seek a particular solution y, of the non-homogeneous problem y" + y' 2y = 36e -2r using the method of undetermined coefficients (See the link below for a help sheet) 4) Apply the method of undetermined coefficients to find y, We then find the general solution as a sum the complementary solution yc y = y. + yp- Finally you are asked to use the general solution to solve an IVP, C1Y1 C2Y2 and a particular solution: 5) Given the initial conditions y(0) = -2 and y' (0) = -11 find the unique solution to the IVP
tHe hon-homogeneous problem y" + y' - 2y = 36e-2z First we consider the homogeneous problem y" + y' - 2y = 0: 1) the auxiliary equation is ar?+ br + c= ^2+r-2 = 0. 2) The roots of the auxiliary equation are 1, -2 (enter answers as a comma separated list). 3) A fundamental set of solutions is e^x, e^(-2x) (enter answers as a comma separated list). Using these we obtain the the complementary solution y. = c¡Y1 + c2Y2 for arbitrary constants Cq and c2- Next we seek a particular solution y, of the non-homogeneous problem y" + y' 2y = 36e -2r using the method of undetermined coefficients (See the link below for a help sheet) 4) Apply the method of undetermined coefficients to find y, We then find the general solution as a sum the complementary solution yc y = y. + yp- Finally you are asked to use the general solution to solve an IVP, C1Y1 C2Y2 and a particular solution: 5) Given the initial conditions y(0) = -2 and y' (0) = -11 find the unique solution to the IVP
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Solving a Non-Homogeneous Differential Equation
We consider the non-homogeneous problem:
\[ y'' + y' - 2y = 36e^{-2x} \]
#### Step 1: Solve the Homogeneous Equation
First, consider the corresponding homogeneous problem:
\[ y'' + y' - 2y = 0 \]
1. **Auxiliary Equation**:
\[ ar^2 + br + c = 0 \]
For this problem, the auxiliary equation is:
\[ r^2 + r - 2 = 0 \]
2. **Roots of the Auxiliary Equation**:
The roots are:
\[ 1, -2 \]
3. **Fundamental Set of Solutions**:
A fundamental set of solutions is:
\[ e^x, e^{-2x} \]
The complementary solution \( y_c \) can be written as:
\[ y_c = c_1 y_1 + c_2 y_2 \]
Where \( c_1 \) and \( c_2 \) are arbitrary constants.
#### Step 2: Find a Particular Solution
Next, we seek a particular solution \( y_p \) of the non-homogeneous problem:
\[ y'' + y' - 2y = 36e^{-2x} \]
We use the method of undetermined coefficients.
4. **Apply the Method of Undetermined Coefficients**:
Find \( y_p \) using this method.
#### Step 3: General Solution
The general solution is a sum of the complementary solution \( y_c \) and a particular solution \( y_p \):
\[ y = y_c + y_p \]
Finally, you are asked to use the general solution to solve an initial value problem (IVP).
5. **Initial Conditions**:
Given:
\[ y(0) = -2 \; \text{and} \; y'(0) = -11 \]
Find the unique solution to the IVP:
\[ y = \ldots \]
By following these steps, you can solve the non-homogeneous second-order linear differential equation using the method of undetermined coefficients and initial conditions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbe424177-0c47-437b-8b35-993829812e07%2Feab72c2e-c5bd-48c2-a1f5-448a7be964b7%2Fe4ezn1k_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving a Non-Homogeneous Differential Equation
We consider the non-homogeneous problem:
\[ y'' + y' - 2y = 36e^{-2x} \]
#### Step 1: Solve the Homogeneous Equation
First, consider the corresponding homogeneous problem:
\[ y'' + y' - 2y = 0 \]
1. **Auxiliary Equation**:
\[ ar^2 + br + c = 0 \]
For this problem, the auxiliary equation is:
\[ r^2 + r - 2 = 0 \]
2. **Roots of the Auxiliary Equation**:
The roots are:
\[ 1, -2 \]
3. **Fundamental Set of Solutions**:
A fundamental set of solutions is:
\[ e^x, e^{-2x} \]
The complementary solution \( y_c \) can be written as:
\[ y_c = c_1 y_1 + c_2 y_2 \]
Where \( c_1 \) and \( c_2 \) are arbitrary constants.
#### Step 2: Find a Particular Solution
Next, we seek a particular solution \( y_p \) of the non-homogeneous problem:
\[ y'' + y' - 2y = 36e^{-2x} \]
We use the method of undetermined coefficients.
4. **Apply the Method of Undetermined Coefficients**:
Find \( y_p \) using this method.
#### Step 3: General Solution
The general solution is a sum of the complementary solution \( y_c \) and a particular solution \( y_p \):
\[ y = y_c + y_p \]
Finally, you are asked to use the general solution to solve an initial value problem (IVP).
5. **Initial Conditions**:
Given:
\[ y(0) = -2 \; \text{and} \; y'(0) = -11 \]
Find the unique solution to the IVP:
\[ y = \ldots \]
By following these steps, you can solve the non-homogeneous second-order linear differential equation using the method of undetermined coefficients and initial conditions.
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