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Chapter 6: Problem 37

Finding a Particular Solution In Exercises \(37-42,\) verify that the general solution satisfies the differential equation. Then find the particular solution that satisfies the initial condition(s). $$\begin{array}{l}{y=C e^{-6 x}} \\ {y^{\prime}+6 y=0} \\ {y=3 \text { when } x=0}\end{array}$$

Short Answer

Expert verified
The particular solution of the differential equation given the initial condition \(y = 3\) when \(x = 0\) is \(y = 3e^{-6x}\).

Step by step solution

01

Verify the General Solution

Start by substituting \(y = Ce^{-6x}\) into the differential equation \(y' + 6y = 0\). The derivative of \(y = Ce^{-6x}\) is \(y' = -6Ce^{-6x}\). Substituting these into the given differential equation gives \(-6Ce^{-6x} + 6Ce^{-6x} = 0\), which simplifies to zero, showing the general solution is correct.
02

Apply the Initial Condition

Given that \(y = 3\) when \(x = 0\), substitute into the general solution to find the value of C. So, \(3 = Ce^{-6*0}\), which simplifies to \(3 = C*1 = C\). Therefore, the constant \(C = 3\).
03

Write Down the Specific Solution

Substitute C into the general solution to get the specific solution. This gives \(y = 3e^{-6x}\).

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Key Concepts

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

Differential Equation
A differential equation is an equation that involves a function and its derivatives. It's a way to describe how a certain quantity changes over time or space. In our exercise, the differential equation given is
  • \( y' + 6y = 0 \)
This equation involves the function \( y \) and its first derivative \( y' \). A key idea in solving differential equations is finding functions that satisfy these conditions, which in our case, tells us how the function \( y \) behaves as \( x \) changes.
It's like finding a rule that matches the pattern described by the equation. Understanding this allows you to predict future outcomes, which is fundamental in fields such as physics, engineering, and biology.
General Solution
The general solution of a differential equation is a formula that contains one or more arbitrary constants. These constants can be adjusted to fit particular situations. In our exercise, the general solution is expressed as:
  • \( y = Ce^{-6x} \)
Here, \( C \) represents the arbitrary constant. This constant can vary, which means the general solution provides a family of solutions rather than a unique answer.
This flexibility is crucial because it means the general solution can be adapted to fit different conditions or initial data presented by various problems.
Initial Condition
An initial condition is a piece of extra information provided to help find a specific solution from the general solution. It usually gives a value of the function at a particular point. In this problem, the initial condition stated is:
  • \( y = 3 \) when \( x = 0 \)
This condition allows us to determine the specific value of the arbitrary constant from the general solution. By plugging in this initial value into the general solution, we found:
  • \( 3 = Ce^{0} \)
Simplifying it, results into \( C = 3 \). This is crucial as it gives us the unique particular solution that precisely fits the condition given in the problem.
Exponential Function
The exponential function arises frequently in differential equations due to its unique property - its derivative is proportional to the value of the function itself. In our problem, the solution involves the exponential function:
  • \( y = 3e^{-6x} \)
The term \( e^{-6x} \) is an exponential function. The constant \( e \) is approximately 2.71828, a special number arising in continuous growth or decay problems. Here, a negative exponent
  • \( -6x \)
indicates that the function \( y \) decreases as \( x \) increases, resulting in exponential decay. This behavior is common in natural processes such as radioactive decay, cooling of objects, and more, marking the importance of understanding the exponential function.

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Most popular questions from this chapter

True or False? In Exercises 67 and \(68,\) determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. Half of the atoms in a sample of radioactive radium decay in 799.5 years.

In Exercises 47 and \(48,\) (a) use a graphing utility to graph the slope field for the differential equation, (b) find the particular solutions of the differential equation passing through the given points, and (c) use a graphing utility to graph the particular solutions on the slope field in part (a). \(\begin{array}{ll}{\text { Differential Equation }} & {\text { Points }} \\\ {\frac{d y}{d y}{d x}+4 x^{3} y=x^{3}} & \quad\left(0, \frac{7}{2}\right),\left(0,-\frac{1}{2}\right)\end{array}\)

Modeling Data One hundred bacteria are started in a culture and the number \(N\) of bacteria is counted each hour for 5 hours. The results are shown in the table, where \(t\) is the time in hours. $$\begin{array}{|c|c|c|c|c|c|c|}\hline t & {0} & {1} & {2} & {3} & {4} & {5} \\\ \hline N & {100} & {126} & {151} & {198} & {243} & {297} \\\ \hline\end{array}$$ (a) Use the regression capabilities of a graphing utility to find an exponential model for the data.(b) Use the model to estimate the time required for the population to quadruple in size.

Determining if a Function Is Homogeneous In Exercises \(69-76,\) determine whether the function is homogeneous, and if it is, determine its degree. A function \(f(x, y)\) is homogeneous of degree \(n\) if \(f(x, t y)=t^{n} f(x, y) .\) $$f(x, y)=e^{x / y}$$

Using an Integrating Factor The expression \(u(x)\) is an integrating factor for \(y^{\prime}+P(x) y=Q(x) .\) Which of the following is equal to \(u^{\prime}(x) ?\) Verify your answer. (a) \(P(x) u(x) \quad\) (b) \(P^{\prime}(x) u(x)\) (c) \(Q(x) u(x) \quad\) (d) \(Q^{\prime}(x) u(x)\)
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