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Chapter 9: Problem 12

Verifying general solutions Verify that the given function is a solution of the differential equation that follows it. Assume \(C, C_{1}, C_{2}\) and \(C_{3}\) are arbitrary constants. $$u(t)=C_{1} e^{t}+C_{2} t e^{t} ; u^{\prime \prime}(t)-2 u^{\prime}(t)+u(t)=0$$

Short Answer

Expert verified
Question: Verify that the function u(t) = C1e^t + C2te^t is a solution to the differential equation u''(t) - 2u'(t) + u(t) = 0. Answer: After finding the first and second derivatives of u(t) and substituting them into the differential equation, the expression simplifies to 0. This confirms that the given function u(t) is a solution to the given differential equation.

Step by step solution

01

Find u'(t)

To find the first derivative, u'(t), we will differentiate u(t) with respect to t. Using rules for differentiation, we have: $$u'(t) = \frac{d}{dt} (C_1e^t + C_2te^t) = C_1e^t + C_2e^t + C_2te^t$$
02

Find u''(t)

Now, we will find the second derivative, u''(t), by differentiating u'(t) with respect to t. Using the rules for differentiation again, we have: $$u''(t) = \frac{d}{dt} (C_1e^t + C_2e^t + C_2te^t) = C_1e^t + C_2e^t + C_2e^t + C_2te^t$$
03

Substitute u(t), u'(t), and u''(t) into the differential equation

Now we will substitute the expressions for u(t), u'(t), and u''(t) into the given differential equation: $$u''(t) - 2u'(t) + u(t) = (C_1e^t + C_2e^t + C_2e^t + C_2te^t) - 2(C_1e^t + C_2e^t + C_2te^t) + (C_1e^t + C_2te^t)$$
04

Simplify the expression

Now, we will simplify the expression obtained in the previous step, to see if it amounts to 0. $$C_1e^t + C_2e^t + C_2e^t + C_2te^t - 2C_1e^t - 2C_2e^t - 2C_2te^t + C_1e^t + C_2te^t = 0$$ After simplification, we get: $$0 = 0$$ Since the expression simplifies to 0, this confirms that the given function u(t) is a solution to the given differential equation.

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Key Concepts

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

Verifying Solutions
When working with differential equations, one often comes across the task of verifying whether a given function is indeed a solution. This means checking if the function satisfies the equation when substituted into it. The process involves a few key steps:
  • Understand the form of the differential equation and the solution provided.
  • Use the rules for differentiation to find the derivatives of the proposed solution.
  • Substitute these derivatives along with the solution back into the differential equation.
  • Simplify the resulting expression to see if you arrive at a logical equivalence, often zero equals zero (0=0) in the context of homogeneous equations.
When the process above yields an identity, such as 0=0, this verifies that the proposed function is indeed a solution to the differential equation. This method showcases the interconnected nature of differential equations and calculus, as the verification directly depends on derivative calculations.
Second Derivative
The second derivative of a function is a measure of how the rate of change of the function's rate of change is varying. In layman's terms, while the first derivative represents velocity if the function is position, the second derivative represents acceleration. The process to find the second derivative is as follows:
  • Find the first derivative of the function.
  • Differentiate the first derivative with respect to its variable to obtain the second derivative.
The notation for the second derivative is usually f''(x) if the function is f(x), or y'' if the function is described as y. This concept is crucial in verifying solutions for differential equations because it often informs us about the behavior of solutions such as their concavity, or in the case of physics, the acceleration of a particle in motion.
Rules for Differentiation
In calculus, the rules for differentiation serve as the foundation for solving various problems, including differential equations. These rules include:
  • The Power Rule, which tells us that the derivative of _x^n is nx^{n-1}.
  • The Product Rule, required when differentiating the product of two functions.
  • The Quotient Rule, used when differentiating a quotient of two functions.
  • The Chain Rule, which is vital for differentiating composite functions.
  • The Exponential Rule, especially important when dealing with functions involving the mathematical constant e, such as e^x.
Applying these rules correctly allows students to find derivatives of complex functions that are often parts of differential equations. For example, in the exercise mentioned earlier, the Product and Exponential Rules are particularly useful when differentiating the function u(t) = C_1e^t + C_2te^t.
Differential Equation
A differential equation is an equation that involves a function and its derivatives. These equations describe relationships that involve rates of change and are fundamental in math, engineering, and science. They can be classified based on several factors:
  • The order of the differential equation is determined by the highest derivative present in the equation.
  • Linearity, where a linear differential equation has the dependent variable and all its derivatives to the first power and no products of them.
  • The equation can be homogeneous if it equals zero, or non-homogeneous otherwise.
Solving a differential equation often means finding all functions that satisfy the equation, commonly known as the general solution. In other cases, initial conditions are provided, resulting in a specific solution known as a particular solution.

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Most popular questions from this chapter

Consider the differential equation \(y^{\prime \prime}(t)-k^{2} y(t)=0,\) where \(k>0\) is a real number. a. Verify by substitution that when \(k=1\), a solution of the equation is \(y(t)=C_{1} e^{t}+C_{2} e^{-t} .\) You may assume this function is the general solution. b. Verify by substitution that when \(k=2\), the general solution of the equation is \(y(t)=C_{1} e^{2 t}+C_{2} e^{-2 t}\) c. Give the general solution of the equation for arbitrary \(k>0\) and verify your conjecture. d. For a positive real number \(k\), verify that the general solution of the equation may also be expressed in the form \(y(t)=C_{1} \cosh k t+C_{2} \sinh k t,\) where cosh and sinh are the hyperbolic cosine and hyperbolic sine, respectively (Section \(7.3)\)

Solve the differential equation for Newton's Law of Cooling to find the temperature function in the following cases. Then answer any additional questions. A glass of milk is moved from a refrigerator with a temperature of \(5^{\circ} \mathrm{C}\) to a room with a temperature of \(20^{\circ} \mathrm{C}\). One minute later the milk has warmed to a temperature of \(7^{\circ} \mathrm{C}\). After how many minutes does the milk have a temperature that is \(90 \%\) of the ambient temperature?

In each case, carry out the following steps. a. Identify which equation corresponds to the predator and which corresponds to the prey. b. Find the lines along which \(x^{\prime}(t)=0 .\) Find the lines along which \(y^{\prime}(t)=0\) c. Find the equilibrium points for the system. d. Identify the four regions in the first quadrant of the xy-plane in which \(x^{\prime}\) and \(y^{\prime}\) are positive or negative. e. Sketch a representative solution curve in the xy-plane and indicate the direction in which the solution evolves. $$x^{\prime}(t)=-3 x+6 x y, y^{\prime}(t)=y-4 x y$$

Find the equilibrium solution of the following equations, make a sketch of the direction field, for \(t \geq 0,\) and determine whether the equilibrium solution is stable. The direction field needs to indicate only whether solutions are increasing or decreasing on either side of the equilibrium solution. $$u^{\prime}(t)+7 u+21=0$$

Let \(y(t)\) be the population of a species that is being harvested, for \(t \geq 0 .\) Consider the harvesting model \(y^{\prime}(t)=0.008 y-h, y(0)=y_{0},\) where \(h\) is the annual harvesting rate, \(y_{0}\) is the initial population of the species, and \(t\) is measured in years. a. If \(y_{0}=2000,\) what harvesting rate should be used to maintain a constant population of \(y=2000,\) for \(t \geq 0 ?\) b. If the harvesting rate is \(h=200 /\) year, what initial population ensures a constant population?
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