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Chapter 10: Problem 22

Solve the following differential equations with the given initial conditions. $$y^{\prime}=t^{2} e^{-3 y}, y(0)=2$$

Short Answer

Expert verified
The solution to the differential equation is \( y = \frac{1}{3} \text{ln}(t^3 + e^6) \).

Step by step solution

01

Separate the variables

Rewrite the differential equation to isolate the variables on each side. Start with \[y^{\text{'}} = t^2 e^{-3y}\]. This can be rearranged to \[e^{3y} dy = t^2 dt\].
02

Integrate both sides

Integrate both sides separately. The left side becomes \( \begin{aligned} \frac{1}{3} e^{3y} \text{ and the right side becomes } \frac{t^3}{3} + C.\end{aligned} \)
03

Include the constant of integration

Combine the results of the integration to get \[ \frac{1}{3} e^{3y} = \frac{1}{3} t^3 + C \].
04

Solve for the constant

Use the initial condition to find the constant. Substitute \(t = 0\) and \( y = 2 \) into the equation to solve for \(C\):\[ \frac{1}{3} e^{3 \cdot 2} = \frac{1}{3} \cdot 0^3 + C \]This simplifies to \[ \frac{1}{3} e^6 = C \].
05

Substitute the constant back into the equation

Replace \(C\) in the integrated equation with \(\frac{1}{3} e^6\):\[ \frac{1}{3} e^{3y} = \frac{1}{3} t^3 + \frac{1}{3} e^6 \].Multiply the whole equation by 3 to simplify:\[ e^{3y} = t^3 + e^6 \].
06

Solve for y

Finally, solve for \(y\):Take the natural logarithm of both sides:\[ 3y = \text{ln}(t^3 + e^6) \]Then divide by 3:\[ y = \frac{1}{3} \text{ln}(t^3 + e^6) \].

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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 crucial technique in solving differential equations. It involves rearranging the equation so that all terms involving one variable are on one side of the equation, and all terms involving the other variable are on the opposite side. For instance, if you have a differential equation of the form \( y^{\text{'}}=t^2 e^{-3y} \), you can separate the variables by rewriting it as \(e^{3y} dy = t^2 dt\). Contrarily, if you have mixed variables on both sides, separating variables allows you to integrate both sides independently, making the equation easier to solve.
Initial Conditions
Initial conditions are given values that help in solving differential equations uniquely. These conditions define specific values for the function and its derivatives at a particular point. Taking the initial condition \( y(0) = 2 \) as an example, it tells us that when \( t = 0 \), the value of \( y \) is 2. By applying these conditions, we can find the constant of integration, ensuring the solution fits the given scenario exactly. This step is pivotal as it transforms a general solution into a particular solution that meets the problem's requirements.
Integration
Integration is the process of finding the integral of a function, which is the reverse process of differentiation. When solving differential equations, after separating the variables, you integrate each side to find the general solution. For example, integrating both sides of the separated equation \( e^{3y} dy = t^2 dt \), the result is \( \frac{1}{3} e^{3y} = \frac{t^3}{3} + C \). Integration requires you to remember different integration rules and sometimes to apply techniques like substitution or partial fractions to evaluate the integrals.
Natural Logarithm
The natural logarithm, denoted as \( \text{ln} \), is the inverse function of the exponential function \( e^x \). It plays a crucial role in solving differential equations, especially when you need to isolate variables. For instance, when you have an equation like \( e^{3y} = t^3 + e^6 \), you can apply the natural logarithm to both sides to get \( 3y = \text{ln}(t^3 + e^6) \). This step simplifies the process of solving for \( y \). Using natural logarithms effectively often requires understanding properties such as \( \text{ln}(ab) = \text{ln}(a) + \text{ln}(b) \) and \( \text{ln}(e^x) = x \).

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Most popular questions from this chapter

Solve the following differential equations: $$y y^{\prime}=t \sin \left(t^{2}+1\right)$$

Suppose that the Consumer Products Safety Commission issues new regulations that affect the toy-manufacturing industry. Every toy manufacturer will have to make certain changes in its manufacturing process. Let \(f(t)\) be the fraction of manufacturers that have complied with the regulations within \(t\) months. Note that \(0 \leq f(t) \leq 1 .\) Suppose that the rate at which new companies comply with the regulations is proportional to the fraction of companies who have not yet complied, with constant of proportionality \(k=.1\). (a) Construct a differential equation satisfied by \(f(t).\) (b) Use Euler's method with \(n=3\) to estimate the fraction of companies that comply with the regulations within the first 3 months. (c) Solve the differential equation in part (a) and compute \(f(3).\) (d) Compare the answers in parts (b) and (c) and approximate the error in using Euler's method.

The National Automobile Dealers Association reported that the average retail selling price of a new vehicle was \(\$ 30,303\) in 2012. A person purchased a new car at the average price and financed the entire amount. Suppose that the person can only afford to pay \(\$ 500\) per month. Assume that the payments are made at a continuous annual rate and that interest is compounded continuously at the rate of \(3.5 \%\). (Source: The National Automobile Dealers Association, www.nada.com.) (a) Set up a differential equation that is satisfied by the amount \(f(t)\) of money owed on the car loan at time \(t.\) (b) How long will it take to pay off the car loan?

Probability of Accidents Let \(t\) represent the total number of hours that a truck driver spends during a year driving on a certain highway connecting two cities, and let \(p(t)\) represent the probability that the driver will have at least one accident during these \(t\) hours. Then, \(0 \leq p(t) \leq 1,\) and \(1-p(t)\) represents the probability of not having an accident. Under ordinary conditions, the rate of increase in the probability of an accident (as a function of \(t\) ) is proportional to the probability of not having an accident. Construct and solve a differential equation for this situation.

Review concepts that are important in this section. In each exercise, sketch the graph of a function with the stated properties. Domain: \(0 \leq t \leq 4 ;(0,2)\) is on the graph; the slope is always positive, and the slope becomes more positive (as \(t\) increases).
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