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Chapter 10: Problem 1

Solve the following differential equations: $$ \frac{d y}{d t}=\frac{5-t}{y^{2}} $$

Short Answer

Expert verified
The solution is \( y = \sqrt[3]{3(5t - \frac{t^2}{2} + C)} \).

Step by step solution

01

- Separate the variables

Rewrite the given differential equation by separating the variables. Start by multiplying both sides by \( y^2 \, dt \) to get: \[ y^2 \, dy = (5 - t) \, dt \]
02

- Integrate both sides

Integrate the left side with respect to \( y \) and the right side with respect to \( t \). \[ \ \int y^2 \, dy = \int (5 - t) \, dt \]
03

- Perform the integrations

Calculate the integrals: \[ \int y^2 \, dy = \frac{y^3}{3} + C_1 \] \[ \int (5 - t) \, dt = 5t - \frac{t^2}{2} + C_2 \]
04

- Combine the constants

Combine the constants of integration \( C_1 \) and \( C_2 \) into a single constant \( C \): \[ \frac{y^3}{3} = 5t - \frac{t^2}{2} + C \]
05

- Solve for \( y \)

Multiply both sides by 3 to isolate \( y^3 \): \[ y^3 = 3 \left( 5t - \frac{t^2}{2} + C \right) \]
06

- Express the final solution

Take the cube root of both sides to solve for \( y \): \[ y = \sqrt[3]{3 \left( 5t - \frac{t^2}{2} + C \right)} \]

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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 powerful technique to solve differential equations. It involves rearranging the equation such that each variable appears on opposite sides of the equation. This allows us to integrate each side independently.
In the original problem, we start with the equation: \[ \frac{d y}{d t} = \frac{5 - t}{y^{2}} \]
To separate the variables, we multiply both sides by \( y^2 \, dt \). This step moves all terms involving \( y \) to one side and all terms involving \( t \) to the other side: \[ y^2 \, dy = (5 - t) \, dt \]
Now, we've separated the variables successfully. Each side can now be integrated independently.
Integration
Integration is the inverse operation of differentiation. It allows us to find a function given its derivative.
In the separated form of our equation: \[ y^2 \, dy = (5 - t) \, dt \]
we can now integrate both sides:
    \[ \int y^2 \, dy = \int (5 - t) \, dt \]
The integral of \( y^2 \) with respect to \( y \) is \( \frac{y^3}{3} + C_1 \), and the integral of \( 5 - t \) with respect to \( t \) is \( 5t - \frac{t^2}{2} + C_2 \). These integrals give us the functions whose derivatives are the original expressions.
Constants of Integration
When performing indefinite integration, we add a constant of integration because there are infinitely many antiderivatives for any given function.
In our solutions, these appear as \( C_1 \) and \( C_2 \): \[ \frac{y^3}{3} + C_1 = 5t - \frac{t^2}{2} + C_2 \]
Combining these constants into a single constant \( C \) simplifies the equation:
    \[ \frac{y^3}{3} = 5t - \frac{t^2}{2} + C \]
This new constant \( C \) encompasses both initial constants of integration, allowing for a cleaner and simpler final expression.
Solving Differential Equations
Solving a differential equation involves finding a function \( y(t) \) that satisfies the given equation.
We began with our differential equation, separated the variables, and integrated both sides. Our next step is to solve for \( y \):
    \

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Most popular questions from this chapter

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?

A person took out a loan of $$\$ 100,000$$ from a bank that charges \(7.5 \%\) interest compounded continuously. What should be the annual rate of payments if the loan is to be paid in full in exactly 10 years? (Assume that the payments are made continuously throughout the year.)

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).

Solve the following differential equations with the given initial conditions. $$ \frac{d N}{d t}=2 t N^{2}, N(0)=5 $$

Let \(c\) be the concentration of a solute outside a cell that we assume to be constant throughout the process, that is, unaffected by the small influx of the solute across the membrane due to a difference in concentration. The rate of change of the concentration of the solute inside the cell at any time \(t\) is proportional to the difference between the outside concentration and the inside concentration. Set up the differential equation whose solution is \(y=f(t)\), the concentration of the solute inside the cell at time \(t\). Sketch a solution.
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