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Chapter 2: Problem 37

It is sometimes possible to transform a nonexact differential equation \(M(x, y) d x+N(x, y) d y=0\) into an exact equation by multiplying it by an integrating factor \(\mu(x, y)\). In Problems \(37-42\) solve the given equation by verifying that the indicated function \(\mu(x, y)\) is an integrating factor. $$ 6 x y d x+\left(4 y+9 x^{2}\right) d y=0, \quad \mu(x, y)=y^{2} $$

Short Answer

Expert verified
The solution is \(3x^2y^3 + y^4 = C\).

Step by step solution

01

Verify Non-exactness

To determine whether the given differential equation is non-exact, calculate \(\frac{\partial M}{\partial y}\) and \(\frac{\partial N}{\partial x}\). Here, \(M(x, y) = 6xy\) and \(N(x, y) = 4y + 9x^2\). Compute \(\frac{\partial M}{\partial y} = 6x\) and \(\frac{\partial N}{\partial x} = 18x\). Since \(\frac{\partial M}{\partial y} eq \frac{\partial N}{\partial x}\), the equation is non-exact.
02

Multiply by Integrating Factor

Use the given integrating factor \(\mu(x, y) = y^2\) and multiply the entire differential equation by this factor. This results in:\[ y^2(6xy) \, dx + y^2(4y + 9x^2) \, dy = 0.\]Simplifying, we have:\[ 6xy^3 \, dx + (4y^3 + 9x^2y^2) \, dy = 0.\]
03

Check Exactness of Transformed Equation

For the new equation \(M(x, y) = 6xy^3\) and \(N(x, y) = 4y^3 + 9x^2y^2\), compute \(\frac{\partial M}{\partial y} = 18xy^2\) and \(\frac{\partial N}{\partial x} = 18xy^2\). As \(\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}\), the equation is now exact.
04

Solve the Exact Equation

To solve an exact equation, find a potential function \(\psi(x, y)\) such that \(\frac{\partial \psi}{\partial x} = M\) and \(\frac{\partial \psi}{\partial y} = N\). Integrate \(M = 6xy^3\) with respect to \(x\) to get \(\psi(x, y) = 3x^2y^3 + g(y)\). Now, differentiate \(\psi\) with respect to \(y\) to get \(9x^2y^2 + g'(y) = 4y^3 + 9x^2y^2\). Solve for \(g(y)\) by identifying it as \(g'(y) = 4y^3\), which integrates to \(g(y) = y^4\). Hence, \(\psi(x, y) = 3x^2y^3 + y^4\).
05

Express in Implicit Form

The solution to the differential equation is given by setting the potential function \(\psi(x, y)\) to a constant: \[3x^2y^3 + y^4 = C.\] This expression represents the general solution of the differential equation after applying the integrating factor.

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Key Concepts

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

Integrating Factors
If you come across a non-exact differential equation, you might need to use a special trick to make it work. This trick involves an integrating factor. An integrating factor is like a magical multiplier that, when applied to the original equation, turns it into an exact differential equation. That's really handy because exact equations are much easier to solve.

Let's break down the process:
  • First, check if your equation is non-exact by comparing partial derivatives.
  • If it is non-exact, apply the integrating factor \( \mu(x, y) \) to each term of the equation.
  • Re-evaluate the equation to see if it has become exact.
In our example, multiplying the equation by an integrating factor \( y^2 \) transforms it effectively, leading us to a much simpler path to finding the solution.
Potential Function
When you've successfully transformed a non-exact equation into an exact one, it's time to find the potential function. Think of this function as the central piece to your solution puzzle. A potential function \( \psi(x, y) \) is like a foundation that you build your solution upon.

Here's how it works:
  • Start by integrating \( M(x, y) \) with respect to \( x \). This gives you part of your potential function.
  • Next, differentiate this result with respect to \( y \) and compare with \( N(x, y) \). You might have to find another function of \( y \) to make both sides equal.
  • Integrate this new function of \( y \) to complete your potential function.
In the exercise, after integrating and differentiating, the potential function we find is \( \psi(x, y) = 3x^2y^3 + y^4 \). This expression becomes the solution when it's set to a constant.
Exactness Condition
Determining whether a differential equation is exact is an important step in solving it. The exactness condition tells you whether or not an equation can be easily transformed into a potential function.

Here's how to verify exactness:
  • Calculate the partial derivative of \( M(x, y) \) with respect to \( y \).
  • Calculate the partial derivative of \( N(x, y) \) with respect to \( x \).
  • If these two derivatives are equal, your equation is exact! If not, you need to find an integrating factor.
In the example given, we found that \( \frac{\partial M}{\partial y} = 6x \) and \( \frac{\partial N}{\partial x} = 18x \) were not equal initially, indicating a non-exact equation. However, after applying the integrating factor, the derivatives became equal, achieving exactness.
Non-exact Differential Equations
Not all differential equations start off in a form that's straightforward to solve. These are called non-exact differential equations. Understanding when you have one of these on your hands is crucial. They don't satisfy the condition of exactness on their own.

Usually, with non-exact differential equations:
  • The partial derivatives of \( M(x, y) \) and \( N(x, y) \) do not match, indicating a flaw in symmetry.
  • You cannot directly integrate to find a solution. An integrating factor is imperative to proceed.
In exercises like the one we're discussing, the initial differentials of \( M \) and \( N \) don't meet the required conditions. Hence, applying an integrating factor transforms the equation, allowing a proper solution to emerge.

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Most popular questions from this chapter

Solve the given differential equation by separation of variables. $$ \sec y \frac{d y}{d x}+\sin (x-y)=\sin (x+y) $$

In Problems 1-24 determine whether the given equation is exact. If it is exact, solve it. $$ x \frac{d y}{d x}=2 x e^{x}-y+6 x^{2} $$

Classify each differential equation as separable, exact, linear, homogeneous, or Bernoulli. Some equations may be more than one kind. Do not solve the equations. $$\frac{d y}{d x}=\frac{y^{2}+y}{x^{2}+x}$$

Solve the given Bernoulli equation by using an appropriate substitution. $$ x^{2} \frac{d y}{d x}+y^{2}=x y $$

Classify each differential equation as separable, exact, linear, homogeneous, or Bernoulli. Some equations may be more than one kind. Do not solve the equations. $$(x+1) \frac{d y}{d x}=-y+10$$
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