Problem 1

Question

Show by substitution that if $y_{1}(t)$ and $y_{2}(t)$ are solutions to $y^{\prime \prime}(t)+p y^{\prime}(t)+q y(t)=0$ and each of $C_{1}$ and $C_{2}$ is a number then $$ y(t)=C_{1} y_{1}(t)+C_{2} y_{2}(t) $$ is a solution to $y^{\prime \prime}(t)-p y^{\prime}(t)+q y(t)=0$.

Step-by-Step Solution

Verified

Answer

Yes, by substitution, the combined function $ y(t) = C_1 y_1(t) + C_2 y_2(t) $ is a solution.

1Step 1: Understand Differential Equation and Solution

We need to confirm that the function $ y(t) = C_1 y_1(t) + C_2 y_2(t) $ is a solution to the differential equation $ y''(t) + p y'(t) + q y(t) = 0 $, provided $ y_1(t) $ and $ y_2(t) $ are solutions to this equation.

2Step 2: Calculate Derivatives of y(t)

Compute the first and second derivatives of $ y(t) $: \[y'(t) = C_1 y_1'(t) + C_2 y_2'(t)\] \[y''(t) = C_1 y_1''(t) + C_2 y_2''(t)\]

3Step 3: Substitute Derivatives into the Equation

Substitute $ y(t) $, $ y'(t) $, and $ y''(t) $ into the given differential equation. We have: \[y''(t) + p y'(t) + q y(t) = (C_1 y_1''(t) + C_2 y_2''(t)) + p(C_1 y_1'(t) + C_2 y_2'(t)) + q(C_1 y_1(t) + C_2 y_2(t))\]

4Step 4: Group and Simplify Terms

Rearrange terms based on factors of $ C_1 $ and $ C_2 $: \[= C_1(y_1''(t) + p y_1'(t) + q y_1(t)) + C_2(y_2''(t) + p y_2'(t) + q y_2(t))\]

5Step 5: Show Each Term is Zero

Since $ y_1(t) $ and $ y_2(t) $ are solutions to the equation $ y''(t) + p y'(t) + q y(t) = 0 $, each part in the parentheses is zero: \[C_1(0) + C_2(0) = 0\]

6Step 6: Conclusion

Therefore, since the whole expression becomes zero, $ y(t) $ is indeed a solution to $ y''(t) + p y'(t) + q y(t) = 0 $.

Key Concepts

Linear Combination of SolutionsSecond-Order Differential EquationsSuperposition Principle

Linear Combination of Solutions

When dealing with homogeneous differential equations, one important concept is the linear combination of solutions. Simply put, if you have two or more solutions to a differential equation, you can create new solutions by combining them linearly. In this context, a linear combination means you take the solutions and multiply each by a constant, then add them up together.

For instance, if $y_1(t)$ and $y_2(t)$ are solutions to a differential equation, then $y(t) = C_1 y_1(t) + C_2 y_2(t)$ is also a solution. Here, $C_1$ and $C_2$ are constants that can be any real numbers you choose.

This property is especially useful as it allows for constructing a wide variety of solutions from a given set of basic ones. The idea of using linear combinations is a powerful tool, particularly in the realm of differential equations.

Second-Order Differential Equations

Second-order differential equations refer to equations involving the second derivative of a function. Mathematically, these equations can be written as $y''(t) + p y'(t) + q y(t) = 0$, where $y''(t)$ is the second derivative, $y'(t)$ is the first derivative, and $y(t)$ is the function itself.

These equations are called 'homogeneous' if the result is zero. This type of equation often appears in physical systems, like mechanical vibrations or electrical circuits, where the change rate of something is affected by its current state and its rate of change.

The order of a differential equation corresponds to the highest derivative present.
Homogeneous equations have a zero on the right side, meaning the system is self-contained with no external input.

Understanding these characteristics is crucial when analyzing such systems, as knowing about second-order differential equations empowers you to predict behaviors in the systems they describe.

Superposition Principle

The superposition principle is akin to a secret code that helps unlock the solutions to linear differential equations. It states that if you have multiple solutions to a linear homogeneous differential equation, any linear combination of these solutions will also be a solution.

This principle simplifies dealing with complex differential equations by confirming that you can "superimpose" solutions on one another. Essentially, the principle relies on the linearity of the equation; it remains valid as long as the terms do not become non-linear.

Superposition leads to constructing new, meaningful solutions by combining existing ones.
It helps in building general solutions, especially in understanding how different parts of a system might interact.

This understanding paves the way for advanced methods in solving and integrating differential equations, turning what might initially seem like a puzzle into a predictable pattern.

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