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Chapter 10: Problem 17

Solve the following differential equations: $$y^{\prime}=(y-3)^{2} \ln t$$

Short Answer

Expert verified
y(t) = 3 - \frac{1}{C + \frac{1}{2}(\ln t)^2}

Step by step solution

01

Identify the type of differential equation

The given differential equation is of the form \( y' = (y-3)^2 \ln t \)). It is a first-order, nonlinear ordinary differential equation.
02

Separate the variables

Rearrange the equation to separate the variables y and t. \( \frac{dy}{(y-3)^2} = \ln t \frac{dt}{t} \)
03

Integrate both sides

Now integrate both sides: \( \int \frac{1}{(y-3)^2} dy = \int \ln t \frac{dt}{t} \) The left side integrates to \(-\frac{1}{y-3} \) and the right side integrates to \( \frac{1}{2} (\ln t)^2 \). Don鈥檛 forget to include the constant of integration (C).
04

Write the integrated equation

After integrating, the equation becomes: \( -\frac{1}{y-3} = \frac{1}{2}(\ln t)^2 + C \)
05

Solve for y

Rearrange the equation to solve for y: \( -\frac{1}{y-3} - C = \frac{1}{2}(\ln t)^2 \) \( y = 3 - \frac{1}{C + \frac{1}{2}(\ln t)^2} \)
06

Final Solution

The solution to the differential equation is: \( y(t) = 3 - \frac{1}{C + \frac{1}{2}(\ln t)^2} \)

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Key Concepts

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

Ordinary Differential Equations
Ordinary differential equations (ODEs) are equations involving functions of one independent variable and their derivatives.
They are termed 'ordinary' to distinguish from partial differential equations, which involve multiple independent variables.
Our exercise deals with an ODE since it involves derivatives with respect to a single variable, \( t\).
ODEs can describe a wide range of phenomena, from mechanical systems to electrical circuits.
Understanding ODEs is crucial in fields such as engineering, physics, and economics.
They are typically classified into linear and nonlinear, with numerous methods available for their solution.

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Most popular questions from this chapter

Solve the given equation using an integrating factor. Take \(t>0\). $$y^{\prime}=e^{-i}(y+1)$$

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

New Home Prices in 2012 The Federal Housing Finance Board reported that the national average price of a new one-family house in 2012 was \(\$ 278,900 .\) At the same time, the average interest rate on a conventional 30 -year fixed-rate mortgage was \(3.1 \%\). A person purchased a home at the average price, paid a down payment equal to \(10 \%\) of the purchase price, and financed the remaining balance with a 30 -year fixed-rate mortgage. Assume that the person makes payments continuously at a constant annual rate \(A\) and that interest is compounded continuously at the rate of \(3.1 \%\). (Source: The Federal Housing Finance Board, www.fhfb.gov.) (a) Set up a differential equation that is satisfied by the amount \(f(t)\) of money owed on the mortgage at time \(t.\) (b) Determine \(A,\) the rate of annual payments that are required to pay off the loan in 30 years. What will the monthly payments be? (c) Determine the total interest paid during the 30 -year term mortgage.

Let \(f(t)\) be the solution of \(y^{\prime}=-(t+1) y^{2}, y(0)=1 .\) Use Euler's method with \(n=5\) to estimate \(f(1) .\) Then, solve the differential equation, find an explicit formula for \(f(t),\) and compute \(f(1) .\) How accurate is the estimated value of \(f(1) ?\).

Find an integrating factor for each equation. Take \(t>0\). $$y^{\prime}+\sqrt{t} y=2(t+1)$$
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