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Chapter 10: Problem 13

Solve the given equation using an integrating factor. Take \(t>0\). $$ y^{\prime}+\frac{y}{10+t}=0 $$

Short Answer

Expert verified
The solution is \( y = \frac{C}{10+t} \).

Step by step solution

01

Identify the differential equation

The given equation is \[ y^{\text{'}} + \frac{y}{10+t} = 0. \]
02

Rewrite the equation in standard form

Rewrite the differential equation in the form \[ y^{\text{'}} + p(t)y = 0 \] where \[ p(t) = \frac{1}{10+t}. \]
03

Identify the integrating factor

The integrating factor \( \mu(t) \) is given by \[ \mu(t) = e^{\int p(t) dt} = e^{\int \frac{1}{10+t} dt}. \]
04

Compute the integral for the integrating factor

Compute the integral \[ \int \frac{1}{10+t} dt = \ln|10+t|. \] Therefore, \[ \mu(t) = e^{\ln|10+t|} = |10+t| \]. Given that \( t > 0 \), \( |10+t| = 10+t \). Thus, \[ \mu(t) = 10+t. \]
05

Multiply the differential equation by the integrating factor

Multiply both sides of the differential equation by \( 10+t \), resulting in \[ (10+t)y^{\text{'}} + y = 0. \]
06

Rewrite the left-hand side as a derivative

Notice that the left-hand side can be rewritten as the derivative of a product: \[ \frac{d}{dt}[(10+t)y] = 0. \]
07

Integrate both sides with respect to \( t \)

Integrate both sides to get \[ (10+t)y = C \] where \( C \) is the constant of integration.
08

Solve for \( y \)

Solve for \( y \) by dividing both sides by \( 10+t \), resulting in \[ y = \frac{C}{10+t}. \]

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Key Concepts

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

Integrating Factor
In solving first-order linear differential equations, the integrating factor is a powerful tool. It transforms a differential equation into an easily solvable form. For a differential equation of the form: \ \ \( y' + p(t)y = 0 \) The integrating factor \( \mu(t) \) is calculated as: \ \ \( \mu(t) = e^{\int p(t) dt} \) \ \ In the given exercise, \( p(t) = \frac{1}{10+t} \). So the integrating factor becomes: \ \( \mu(t) = e^{\int \frac{1}{10+t} dt} = e^{\ln|10+t|} = |10+t| = 10+t \). \ \ Therefore, the integrating factor simplifies the original equation and helps in finding the solution more directly.
Homogeneous Differential Equation

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Most popular questions from this chapter

When a red-hot steel rod is plunged in a bath of water that is kept at a constant temperature \(10^{\circ} \mathrm{C}\), the temperature of the rod at time \(t\), \(f(t)\), satisfies the differential equation $$ y^{\prime}=k[10-y] $$ where \(k>0\) is a constant of proportionality. Determine \(f(t)\) if the initial temperature of the rod is \(f(0)=350^{\circ} \mathrm{C}\) and \(k=.1\).

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

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