1) Solve The IVP: Y"-9y'+18y=0; Y(0)=1; Y'(0)=-6 2) Determine The Form Of The Particular Solution For
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Introduction
Solving initial value problems (IVPs) involving second-order linear differential equations is a fundamental aspect of differential equations. These problems often arise in physics, engineering, and other scientific disciplines to model systems such as mechanical vibrations, electrical circuits, and thermal processes. In this article, we will explore how to solve the specific IVP:
\[
Y'' - 9Y' + 18Y = 0, \quad Y(0) = 1, \quad Y'(0) = -6
\]
and how to determine the form of the particular solution when dealing with nonhomogeneous equations. This comprehensive guide will walk you through the step-by-step solution process, including the characteristic equation, general solution, applying initial conditions, and understanding the structure of particular solutions.
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Solving the Homogeneous Differential Equation
Step 1: Write the Differential Equation
The differential equation provided is:
\[
Y'' - 9Y' + 18Y = 0
\]
This is a second-order linear homogeneous differential equation with constant coefficients.
Step 2: Find the Characteristic Equation
To solve this, we assume a solution of the form \(Y = e^{rt}\), where \(r\) is a constant to be determined. Substituting into the differential equation yields:
\[
r^2 e^{rt} - 9r e^{rt} + 18 e^{rt} = 0
\]
Dividing through by \(e^{rt}\) (which is never zero):
\[
r^2 - 9r + 18 = 0
\]
This quadratic equation is called the characteristic equation.
Step 3: Solve the Characteristic Equation
The quadratic can be factored or solved using the quadratic formula:
\[
r = \frac{9 \pm \sqrt{(-9)^2 - 4 \times 1 \times 18}}{2}
\]
Calculate the discriminant:
\[
\Delta = 81 - 72 = 9
\]
Since \(\Delta = 9\), the roots are real and distinct:
\[
r = \frac{9 \pm \sqrt{9}}{2} = \frac{9 \pm 3}{2}
\]
Thus, the roots are:
\[
r1 = \frac{9 + 3}{2} = 6, \quad r2 = \frac{9 - 3}{2} = 3
\]
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General Solution to the Homogeneous Equation
With the roots \(r1 = 6\) and \(r2 = 3\), the general solution is:
\[
Y(t) = C1 e^{6t} + C2 e^{3t}
\]
where \(C1\) and \(C2\) are arbitrary constants determined by initial conditions.
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Applying Initial Conditions
Step 1: Find Derivative \(Y'\)
Differentiate the general solution:
\[
Y'(t) = 6 C1 e^{6t} + 3 C2 e^{3t}
\]
Step 2: Use the Initial Conditions
Given:
\[
Y(0) = 1, \quad Y'(0) = -6
\]
Plug in \(t=0\):
\[
Y(0) = C1 e^{0} + C2 e^{0} = C1 + C2 = 1
\]
\[
Y'(0) = 6 C1 e^{0} + 3 C2 e^{0} = 6 C1 + 3 C2 = -6
\]
Step 3: Solve for \(C1\) and \(C2\)
From the first equation:
\[
C1 + C2 = 1 \quad \Rightarrow \quad C2 = 1 - C1
\]
Substitute into the second:
\[
6 C1 + 3 (1 - C1) = -6
\]
\[
6 C1 + 3 - 3 C1 = -6
\]
\[
(6 C1 - 3 C1) + 3 = -6
\]
\[
3 C_1 + 3 = -6
\]
\[
3 C_1 = -9
\]
\[
C_1 = -3
\]
Now, find \(C_2\):
\[
C_2 = 1 - (-3) = 4
\]
Final Solution to the IVP
\[
\boxed{
Y(t) = -3 e^{6t} + 4 e^{3t}
}
\]
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Determining the Form of the Particular Solution
When dealing with nonhomogeneous differential equations, the method of undetermined coefficients is often employed. The form of the particular solution depends on the right-hand side of the differential equation.
Step 1: Identify the Nonhomogeneous Term
Suppose the differential equation is:
\[
Y'' + p(t) Y' + q(t) Y = g(t)
\]
where \(g(t)\) is a known function (e.g., polynomial, exponential, sine, cosine, or a combination).
Step 2: Analyze the Forcing Function \(g(t)\)
Common types of forcing functions and their corresponding particular solutions:
| Forcing Function \(g(t)\) | Typical Form of Particular Solution | Notes |
|-------------------------|--------------------------------------|--------|
| Polynomial (e.g., \(A t^n + \dots\)) | Polynomial of same degree | Adjust if solution overlaps with homogeneous solution |
| Exponential (e.g., \(A e^{k t}\)) | \(A e^{k t}\) | Multiply by \(t\) if \(e^{k t}\) is a solution to homogeneous equation |
| Sine or Cosine (e.g., \(A \sin \omega t + B \cos \omega t\)) | \(A \sin \omega t + B \cos \omega t\) | Multiply by \(t\) if resonance occurs |
Step 3: Formulate the Particular Solution
Based on the form of \(g(t)\), choose an appropriate trial solution with undetermined coefficients. For example:
- For polynomial \(g(t)\), try a polynomial.
- For exponential \(g(t)\), try an exponential multiplied by a polynomial.
- For sinusoidal \(g(t)\), try sinusoidal functions with undetermined coefficients.
Step 4: Adjust for Resonance
If the forcing function is similar to the homogeneous solution, multiply the trial particular solution by \(t\) (or higher powers of \(t\)) to find a suitable form.
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Summary and Key Takeaways
- Homogeneous solutions are found by solving the characteristic equation associated with the differential equation.
- Initial conditions allow for solving arbitrary constants to find a particular solution that fits the problem.
- The general solution of a second-order linear homogeneous differential equation with constant coefficients is a linear combination of exponential functions based on the roots of the characteristic equation.
- When dealing with nonhomogeneous equations, the form of the particular solution depends on the nature of the forcing function \(g(t)\).
- Method of undetermined coefficients is a systematic way to find particular solutions, adjusting the trial form based on resonance conditions.
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Final Remarks
Understanding the process of solving second-order linear differential equations and determining particular solutions is essential for students and professionals working in applied mathematics, physics, and engineering. Properly applying initial conditions ensures the solution accurately models the specific problem scenario. Regular practice with various types of forcing functions enhances mastery of the method of undetermined coefficients and differential equation solving strategies.
If you encounter more complex or non-standard equations, other methods such as variation of parameters or Laplace transforms may be necessary. However, the fundamental principles highlighted in this article serve as a solid foundation for tackling a wide range of differential equations.
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