Find The General Solution To Y+12y+36y=0.
Understanding how to find the general solution of differential equations is fundamental in mathematics, particularly in fields such as physics, engineering, and applied sciences. The given differential equation appears to involve derivatives of a function \( y \), but as it stands, the notation seems to be missing some components. Based on common forms of differential equations, it is likely that the equation was intended to be:
\[
\frac{d^2 y}{dx^2} + 12 \frac{dy}{dx} + 36 y = 0
\]
This is a second-order linear homogeneous differential equation with constant coefficients, which is a classic problem in differential equations. In this article, we will explore how to find its general solution step by step, including the characteristic equation method, roots analysis, and the formulation of the general solution.
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Understanding the Differential Equation
What is the Equation?
The differential equation:
\[
\frac{d^2 y}{dx^2} + 12 \frac{dy}{dx} + 36 y = 0
\]
is a second-order linear homogeneous differential equation with constant coefficients. Such equations are common when modeling physical systems like harmonic oscillators, electrical circuits, and mechanical vibrations.
Why is it Important?
Solving this differential equation provides the general form of solutions that describe the behavior of the system under study. The general solution encompasses all possible solutions, including particular solutions for specific initial conditions.
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Methodology for Solving the Differential Equation
Step 1: Write the Characteristic Equation
For linear differential equations with constant coefficients, the standard approach involves assuming solutions of the form:
\[
y = e^{rx}
\]
Substituting into the differential equation yields the characteristic (or auxiliary) equation:
\[
r^2 + 12 r + 36 = 0
\]
---
Step 2: Solve the Characteristic Equation
This quadratic equation can be solved using factoring, completing the square, or the quadratic formula.
Quadratic Formula:
\[
r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a=1 \), \( b=12 \), and \( c=36 \).
Calculating discriminant:
\[
\Delta = b^2 - 4ac = 12^2 - 4 \times 1 \times 36 = 144 - 144 = 0
\]
Since the discriminant is zero, the roots are real and equal.
Calculating roots:
\[
r = \frac{-12 \pm \sqrt{0}}{2} = \frac{-12}{2} = -6
\]
Thus, the characteristic equation has a repeated root:
\[
r = -6
\]
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Step 3: Write the General Solution
For a second-order linear differential equation with constant coefficients, the solution depends on the nature of the roots:
- Distinct real roots: \( y = C1 e^{r1 x} + C2 e^{r2 x} \)
- Repeated roots: \( y = (A + Bx) e^{r x} \)
- Complex roots: \( y = e^{\alpha x} (C1 \cos \beta x + C2 \sin \beta x) \)
In our case, since there is a repeated real root \( r = -6 \), the general solution takes the form:
\[
\boxed{
y(x) = (A + Bx) e^{-6x}
}
\]
where \( A \) and \( B \) are arbitrary constants determined by initial conditions.
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Summary of the Solution
The process to find the general solution involves recognizing the form of the differential equation, deriving the characteristic equation, solving for roots, and then constructing the solution based on the root types. For the given differential equation, the roots are repeated and real, leading to the general solution:
\[
\boxed{
\boxed{
y(x) = (A + Bx) e^{-6x}
}
}
\]
where \( A \) and \( B \) are arbitrary constants.
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Additional Concepts and Clarifications
Initial Conditions and Particular Solutions
The general solution contains arbitrary constants \( A \) and \( B \), which can be determined if initial or boundary conditions are provided. For example, if \( y(0) \) and \( y'(0) \) are known, then:
\[
\begin{cases}
y(0) = (A + 0) e^{0} = A \\
y'(x) = \frac{d}{dx} [(A + Bx) e^{-6x}]
\end{cases}
\]
Using these, you can solve for \( A \) and \( B \).
Significance of Repeated Roots
Repeated roots indicate solutions where the exponential solution overlaps, requiring the introduction of a multiplicative \( x \) term to find the second linearly independent solution. This is a standard procedure in solving linear differential equations.
Visualizing the Solution
The solution \( y(x) = (A + Bx) e^{-6x} \) describes a damped exponential behavior, which is typical in systems experiencing damping proportional to velocity, such as a mass-spring system with damping.
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Conclusion
The process of solving the differential equation \( \frac{d^2 y}{dx^2} + 12 \frac{dy}{dx} + 36 y = 0 \) demonstrates the power of characteristic equations and roots analysis in understanding the behavior of systems modeled by differential equations. Recognizing the form of the roots—whether real, repeated, or complex—guides the construction of the general solution. In the case of repeated roots, as with our example, the solution involves an exponential multiplied by a polynomial of degree one, leading to:
\[
\boxed{
y(x) = (A + Bx) e^{-6x}
}
\]
This comprehensive approach underscores the essential techniques in solving second-order linear homogeneous differential equations with constant coefficients, equipping students and professionals with the tools to analyze a wide range of dynamic systems.
