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Chapter 7: Problem 2

Transform the given equation into a system of first order equations. \(u^{\prime \prime}+0.5 u^{\prime}+2 u=3 \sin t\)

Short Answer

Expert verified
Question: Transform the given second-order differential equation into a system of first-order differential equations: \(u''+0.5u'+2u=3\sin t\) Answer: The system of first-order differential equations is: 1. \(u' = v\) 2. \(v' = -0.5v - 2u + 3\sin t\)

Step by step solution

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01

Introduce new variable for the first derivative of u

Let \(v = u'\), then \(v' = u''\). Now we can rewrite the given equation in terms of \(u\) and \(v\).
02

Rewrite the equation using the new variables

Replace \(u''\) and \(u'\) with \(v'\) and \(v\) respectively: \(v'+0.5v+2u=3\sin t\)
03

Write the system of first-order equations

We now have two first-order equations: 1. \(u' = v\) 2. \(v' = -0.5v -2u + 3\sin t\) This system of first-order differential equations represents the original second-order equation.

Key Concepts

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

System of Differential Equations
A system of differential equations consists of multiple equations involving derivatives of more than one function. Each equation in the system can have derivatives of one or more dependent variables. By solving the system, you can find specific functions that satisfy all the equations simultaneously. In the original exercise, we started with a second-order differential equation and converted it into a system of first-order equations to make it easier to solve. This method allows you to break down complex relationships into simpler parts.
  • Each equation in the system can describe a different aspect of the overall problem.
  • Systems can be linear or nonlinear, and they can be solved using various mathematical techniques depending on their nature.
Breaking down higher-order equations into systems of first-order equations is a common approach because it simplifies the analysis and computation.
Variable Substitution
Variable substitution is a mathematical technique that involves replacing a variable in an equation with a new variable or expression. This can simplify the equation or transform it into a different form that is easier to solve. In this context, transforming the original second-order differential equation involved using variable substitution. By introducing a new variable, we effectively reduced the complexity of the equation.
  • We replaced the first derivative of the function with a new variable, thus transforming the equation.
  • This change allows us to view the problem from a new perspective, often making it easier to handle.
The strategic use of substitution is crucial in situations where direct solutions to differential equations are not feasible.
Second-Order Differential Equations
Second-order differential equations involve the second derivative of a function. These equations are common in physics and engineering because they frequently describe systems where acceleration, or the rate of change of a rate, is important. Understanding and solving these equations is key in modeling real-world phenomena. In our exercise, the equation\( u'' + 0.5u' + 2u = 3\sin t \) is an example of a second-order differential equation. Such equations can often be challenging to solve directly.
  • They involve terms up to the second derivative, hence more complex than first-order equations.
  • Translating them into a set of first-order equations can make them easier to analyze and solve.
Converting to a system of first-order equations is a powerful tool in dealing with second-order cases, allowing for the application of systematic methods applicable to first-order systems.

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Most popular questions from this chapter

The coefficient matrix contains a parameter \(\alpha\). In each of these problems: (a) Determine the eigervalues in terms of \(\alpha\). (b) Find the critical value or values of \(\alpha\) where the qualitative nature of the phase portrait for the system changes. (c) Draw a phase portrait for a value of \(\alpha\) slightly below, and for another value slightly above, each crititical value. $$ x^{\prime}=\left(\begin{array}{cc}{4} & {\alpha} \\ {8} & {-6}\end{array}\right) \mathbf{x} $$

find a fundamental matrix for the given system of equations. In each case also find the fundamental matrix \(\mathbf{\Phi}(t)\) satisfying \(\Phi(0)=\mathbf{1}\) $$ \mathbf{x}^{\prime}=\left(\begin{array}{rrr}{1} & {-1} & {4} \\ {3} & {2} & {-1} \\ {2} & {1} & {-1}\end{array}\right) \mathbf{x} $$

Express the general solution of the given system of equations in terms of real-valued functions. In each of Problems 1 through 6 also draw a direction field, sketch a few of the trajectories, and describe the behavior of the solutions as \(t \rightarrow \infty\). $$ \mathbf{x}^{\prime}=\left(\begin{array}{ll}{1} & {-1} \\ {5} & {-3}\end{array}\right) \mathbf{x} $$

In each of Problems 15 through 18 solve the given initial value problem. Describe the behavior of the solution as \(t \rightarrow \infty\). $$ \mathbf{x}^{\prime}=\left(\begin{array}{rr}{5} & {-1} \\ {3} & {1}\end{array}\right) \mathbf{x}, \quad \mathbf{x}(0)=\left(\begin{array}{r}{2} \\ {-1}\end{array}\right) $$

The electric circuit shown in Figure 7.6 .6 is described by the system of differential equations \(\frac{d}{d t}\left(\begin{array}{l}{I} \\\ {V}\end{array}\right)=\left(\begin{array}{cc}{0} & {\frac{1}{L}} \\\ {-\frac{1}{C}} & {-\frac{1}{R C}}\end{array}\right)\left(\begin{array}{l}{I} \\\ {V}\end{array}\right)\) where \(I\) is the current through the inductor and \(V\) is the voltage drop across the capacitor. These differential equations were derived in Problem 18 of Section \(7.1 .\) (a) Show that the eigenvalues of the coefficient matrix are real and different if \(L>4 R^{2} C\); show they are complex conjugates if \(L<4 R^{2} C .\) (b) Suppose that \(R=1\) ohm, \(C=\frac{1}{2}\) farad, and \(L=1\) henry. Find the general solution of the system (i) in this case. (c) Find \(I(t)\) and \(V(t)\) if \(I(0)=2\) amperes and \(V(0)=1\) volt (d) For the circuit of part (b) determine the limiting values of \(I(t)\) and \(V(t)\) as \(t \rightarrow \infty\) Do these limiting values depend on the initial conditions?
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