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Chapter 10: Problem 6

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

Short Answer

Expert verified
The integrating factor is \( e^{ -\frac{t^3}{3} } \).

Step by step solution

01

Recognize the standard form of the differential equation

The given differential equation is: \[y^{\text{'} }=t^{2}(y+1)\]Rewriting it in standard linear form: \[y^{\text{'} } - t^{2}y = t^{2}\]
02

Identify the function to integrate

In the standard form \[y^{\text{'} } + P(t) y = Q(t)\], identify \[P(t)\] as \[-t^{2}\]. Therefore, \[P(t) = -t^{2}\]
03

Calculate the integrating factor

The formula for the integrating factor \( \text{IF} \) is \[ \text{IF} = e^{ \int P(t) dt }\]Substitute \[P(t) = -t^2\]: \[ \text{IF} = e^{ \int -t^2 dt }\]Compute the integral: \[ \int -t^2 dt = -\frac{t^3}{3}\]Therefore, the integrating factor is: \[ \text{IF} = e^{ -\frac{t^3}{3} } \]

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Key Concepts

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

Differential Equations
Differential equations involve equations with functions and their derivatives. They're used to model various phenomena in fields like physics, engineering, and economics. For instance, when a rate of change depends on another variable, a differential equation often describes this relationship.
In this exercise, our differential equation is given by: y^{\text{'}} = t^{2}(y+1)
The prime notation \( y^{\text{'}} \) represents the derivative of y with respect to t. We want to solve for y in terms of t.
Linear Form
To use the integrating factor method, we need the differential equation in a standard linear form. The standard linear form for a first-order differential equation is: y^{\text{'} } + P(t)y = Q(t)
Here, we see our given equation: y^{\text{'}} = t^{2}(y+1)
We can rewrite this as: y^{\text{'}} - t^{2}y = t^{2}
In this form, we recognize \( P(t) = -t^{2} \) and \( Q(t) = t^{2} \).
Integral Computation
To find the integrating factor, we need to compute an integral involving \( P(t) \). The integrating factor IF is given by: \text{IF} = e^{ \int P(t) dt }
From the linear form of our equation, we know \( P(t) = -t^{2} \), so we need to calculate: \text{IF} = e^{ \int -t^{2} dt }
Calculating the integral, we get: \( \int -t^{2} dt = -\frac{t^{3}}{3} \)
Therefore, the integrating factor becomes: \text{IF} = e^{ -\frac{t^{3}}{3} } .
Integrating Factor Method
Once we have the integrating factor, the integrating factor method allows us to simplify and solve the differential equation. Multiply every term in the standard form equation by the integrating factor IF: \( e^{ -\frac{t^{3}}{3} }(y^{\text{'} } - t^{2}y) = e^{ -\frac{t^{3}}{3} }(t^{2}) \)
This often transforms the left-hand side into a product rule derivative. The equation then becomes easier to integrate, and we can solve for y.
The integrating factor method is powerful for first-order linear differential equations. It transforms a non-integrable equation into one that is solvable by simple integration, making it a valuable technique in mathematics.

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Most popular questions from this chapter

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

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