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Chapter 31: Problem 22

Show that the given equation is a solution of the given differential equation. $$y^{\prime \prime}+4 y=10 e^{x}, \quad y=c_{1} \sin 2 x+c_{2} \cos 2 x+2 e^{x}$$

Short Answer

Expert verified
Yes, the function is a solution to the differential equation.

Step by step solution

01

Differentiate the solution equation

Start by differentiating the given solution equation, \( y = c_1 \sin 2x + c_2 \cos 2x + 2e^x \). The first derivative is given by \( y' = 2c_1 \cos 2x - 2c_2 \sin 2x + 2e^x \).
02

Find the second derivative

Differentiate the first derivative equation to find the second derivative. This gives \( y'' = -4c_1 \sin 2x - 4c_2 \cos 2x + 2e^x \).
03

Substitute in the differential equation

Substitute \( y'' \) and \( y \) into the differential equation \( y'' + 4y = 10e^x \). This becomes \(-4c_1 \sin 2x - 4c_2 \cos 2x + 2e^x + 4(c_1 \sin 2x + c_2 \cos 2x + 2e^x) = 10e^x \).
04

Simplify the equation

Simplify the equation by combining like terms: \(-4c_1 \sin 2x - 4c_2 \cos 2x + 2e^x + 4c_1 \sin 2x + 4c_2 \cos 2x + 8e^x = 10e^x \). This simplifies further to \(10e^x = 10e^x \).
05

Verify the solution

Since \(10e^x = 10e^x\) is always true, the given function is indeed a solution to the differential equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solution Verification
Solution verification is a crucial step in differential equations since it ensures that the function actually satisfies the given differential equation. When verifying solutions:
  • Start by understanding both the solution and the differential equation.
  • Compute any required derivatives of the solution.
  • Substitute these derivatives back into the original differential equation.
  • Simplify, and check if both sides are equal.
If equality holds, then the function is indeed a correct solution to the differential equation. In our example, through the process of differentiation and substitution, we concluded that the given solution validated the equation as both sides equalized perfectly.
Second Derivative
The second derivative is a measure of how a function's rate of change is itself changing. It's a key concept when working with differential equations.
  • The second derivative, denoted as \( y'' \), is found by differentiating the first derivative \( y' \).
  • It provides insight into the curvature of a graph related to the function.
  • In our example, the function to be differentiated is \( y= c_1 \sin 2x + c_2 \cos 2x + 2e^x \).
  • Its second derivative is \( y'' = -4c_1 \sin 2x - 4c_2 \cos 2x + 2e^x \).
Notice how the trigonometric and exponential parts each contribute differently to the derivative. This helps bind the structure of the differential equation.
Exponential Function
Exponential functions are seen in various types of real-world phenomena. They are also common in differential equations.
  • An exponential function typically has the form \( e^x \), where \( e \) is Euler's number.
  • They have unique properties such as their derivative being equal to themselves.
  • In the example solution, \( 2e^x \) is both part of the solution and the differential equation \( y'' + 4y = 10e^x \).
Exponential functions grow smoothly and continuously, making them an integral part of solving many differential equations, as demonstrated by their presence in the given equation.
Trigonometric Functions
Trigonometric functions often appear in differential equations, particularly in those involving oscillatory or periodic behavior.
  • Functions such as \( \sin x \) and \( \cos x \) describe wave-like patterns.
  • These functions have derivatives that are cyclic, meaning they follow predictable patterns.
  • In our context, the solution contains both \( \sin 2x \) and \( \cos 2x \).
  • Their derivatives, crucial for solution verification, demonstrate how each term behaves with respect to \( x \).
Understanding these functions help break down and analyze differential equations, as they introduce cyclic behaviors that match the nature of the problem.

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Most popular questions from this chapter

Solve the given problems. Explain why more than one function can be a solution of a given differential equation. Find three different functions that are solutions of \(y^{\prime}=y\).

Solve the given problems. Find the equation relating the charge and the time in an electric circuit with the following elements: \(L=0.200 \mathrm{H}, R=8.00 \Omega\) \(C=1.00 \mu \mathrm{F},\) and \(E=0 .\) In this circuit, \(q=0\) and \(i=0.500 \mathrm{A}\) when \(t=0\)

Find the particular solution of each differential equation for the given conditions. $$3 y^{\prime \prime}-10 y^{\prime}+3 y=x e^{-2 x} ; \quad y^{\prime}=-\frac{9}{35} \text { and } y=-\frac{13}{35} \text { when } x=0$$

Solve the given problems. In a given electric circuit \(L=8.00 \mathrm{mH}, R=0, C=0.500 \mu \mathrm{F}\) and \(E=20.0 e^{-200 t} \mathrm{mV}\). Find the relation between the current and the time if \(q=0\) and \(i=0\) for \(t=0\)

Solve the given problems. The displacement \(y\) (in \(\mathrm{cm}\) ) of an object at the end of a spring is described by the equation \(d^{2} y / d t^{2}+4 d y / d t+4 y=0,\) where \(t\) is the time (in s). Find \(y=f(t)\) if \(f(0)=0\) and \(f(1)=0.50 \mathrm{cm}\)
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