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Chapter 7: Problem 2

Solve the differential equation. $$\frac { d y } { d x } = x e ^ { - y }$$

Short Answer

Expert verified
The solution is \( y = \ln\left( \frac{x^{2}}{2} + C \right) \).

Step by step solution

01

Separation of Variables

The given differential equation is \( \frac{dy}{dx} = xe^{-y} \). We can use the method of separation of variables to solve it. Rearrange terms to get \( e^{y} \, dy = x \, dx \). This step separates the variables \( y \) and \( x \) on opposite sides of the equation.
02

Integrate Both Sides

Integrate both sides of the separated equation. The left side becomes \( \int e^{y} \, dy \) and the right side becomes \( \int x \, dx \). This results in \( e^{y} = \frac{x^{2}}{2} + C \), where \( C \) is the constant of integration.
03

Solve for y

To express \( y \) in terms of \( x \), take the natural logarithm of both sides of the equation from Step 2: \( y = \ln\left( \frac{x^{2}}{2} + C \right) \). This provides the 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.

Separation of Variables
Separation of variables is a technique used to solve differential equations. This method involves rearranging an equation so that each of the two variables are on opposite sides of the equation. When you encounter a differential equation, such as \(\frac{dy}{dx} = xe^{-y}\), the goal is to manipulate it into a form where you can independently integrate both sides.
Here's how it works:
  • Move all terms involving \(y\) to one side and all terms involving \(x\) to the other side.
  • The equation then becomes \(e^{y} \, dy = x \, dx\).
This separation is possible because these equations are set up such that each side can be integrated independently, which is a key step in solving the equation.
Once the variables are properly separated, you can proceed to the next step, which is integration.
Integration
Integration is the process of finding the integral, which you can think of as the reverse of taking a derivative. In the context of differential equations, once you've performed separation of variables, you need to integrate both sides with respect to their respective variables.
For our separated equation \(e^{y} \, dy = x \, dx\):
  • The left side \(\int e^{y} \, dy\) simplifies to \(e^{y}\).
  • The right side \(\int x \, dx\) simplifies to \(\frac{x^{2}}{2} + C\).
The result of this integration process is \(e^{y} = \frac{x^{2}}{2} + C\).
Here, the integration constant \(C\) arises, which we will discuss next.
Constant of Integration
The constant of integration, denoted as \(C\), is an important part of the integration process in solving differential equations. Whenever you integrate a function, there is an infinite number of possible solutions that differ by a constant. This constant accounts for those differences.
After integrating both sides of the equation, the result includes this constant: \(e^{y} = \frac{x^{2}}{2} + C\).
Why is it important?
  • It ensures that the solution accounts for all possible shifts or translations that might occur in the family of solutions.
  • It is essential for specifying particular solutions when initial conditions are provided to fully determine \(C\).
In solving the equation, \(C\) allows us to express a general family of curves, and when combined with additional information or conditions, it helps determine the specific curve or solution you're interested in.

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Most popular questions from this chapter

(a) Use a computer algebra system to draw a direction field for the differential equation. Get a printout and use it to sketch some solution curves without solving the differential equation. (b) Solve the differential equation. (c) Use the CAS to draw several members of the family of solutions obtained in part (b). Compare with the curves from part (a) \(y ^ { \prime } = x y\)

Use a computer algebra system to draw a direction field for the given differential equation. Get a printout and sketch on it the solution curve that passes through \((0,1) .\) Then use the CAS to draw the solution curve and compare it with your sketch. \(y^{\prime}=x\left(y^{2}-4\right)\)

Predator-prey equations For each predator-prey system, determine which of the variables, \(x\) or \(y\) , represents the prey population and which represents the predator population. Is the growth of the prey restricted just by the predators or by other factors as well? Do the predators feed only on the prey or do they have additional food sources? Explain. $$\begin{aligned} \text { (a) } \frac{d x}{d t} &=-0.05 x+0.0001 x y \\\ \frac{d y}{d t} &=0.1 y-0.005 x y \\ \text { (b) } \frac{d x}{d t} &=0.2 x-0.0002 x^{2}-0.006 x y \\ \frac{d y}{d t} &=-0.015 y+0.00008 x y \end{aligned}$$

mRNA transcription The intermediate molecule mRNA arises in the decoding of DNA: it is produced by a process called transcription and it eventually decays. Suppose that the rate of transcription is changing exponentially according to the expression \(e ^ { b t } ,\) where \(b\) is a positive constant and mRNA has a constant per capita decay rate of \(k\) . The number of mRNA transcript molecules \(T\) thus changes as \(\frac { d T } { d t } = e ^ { b t } - k T\) Although the form of this equation is similar to that from Exercise \(43 ,\) the first term on the right side is now time varying. As a result, the differential equation is no longer separable; however, the equation can be solved using the change of variables \(y ( t ) = e ^ { k t } T ( t ) .\) Solve the differential equation using this technique.

The Kermack-McKendrick equations are first-order differential equations describing an infectious disease outbreak. Using \(S\) and \(I\) to denote the number of susceptible and infected people in a population, the equations are \(S^{\prime}=-\beta S I \quad I^{\prime}=\beta S I-\mu I\) where \(\beta\) and \(\mu\) are positive constants representing the transmission rate and rate of recovery, respectively. (a) Provide a biological explanation for each term of the equations. (b) Suppose \(\beta=1\) and \(\mu=5 .\) Construct the phase plane including all nullclines, equilibria, and arrows indicating the direction of movement in the plane. (c) Construct the phase plane for arbitrary values of \(\beta\) and \(\mu\) , including all nullclines, equilibria, and arrows indicating direction of movement in the plane.
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