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Magazine Chapter 2: Problem 40
Consider the concept of an integrating factor used in Problems \(29-38 .\) Are the two equations \(M d x+N d y=0\) and \(\mu M d x+\mu N d y=0\) necessarily equivalent in the sense that a solution of one is also a solution of the other? Discuss.
Short Answer
Expert verified
Yes, they are equivalent if \(\mu \neq 0 \).
Step by step solution
01
Understanding the Problem
We are given two differential equations: \( M\,dx + N\,dy = 0 \) and \( \mu M\,dx + \mu N\,dy = 0 \). We want to determine if solving one equation implies solving the other. The idea is that applying an integrating factor \( \mu \) should not change the set of solutions of the equation.
02
Introducing the Integrating Factor
An integrating factor \( \mu \) is a function used to make a non-exact differential equation exact. This means that \( \mu(M\,dx + N\,dy) \) becomes exact, allowing us to find a potential function \( f \) such that \( df = \mu M\,dx + \mu N\,dy = 0 \). If \( \mu \) satisfies this requirement, it is an integrating factor for the equation.
03
Equivalent Solutions
To determine if these equations are equivalent, assume \( (x_0, y_0) \) is a solution to \( M\,dx + N\,dy = 0 \). Then \( x(t) \) and \( y(t) \) satisfy \( M\frac{dx}{dt} + N\frac{dy}{dt} = 0 \). Multiplying through by \( \mu \), we have \( \mu M\frac{dx}{dt} + \mu N\frac{dy}{dt} = 0 \), which suggests \( (x(t), y(t)) \) is also a solution to \( \mu M\,dx + \mu N\,dy = 0 \).
04
Validity of the Integrating Factor
An important condition is that the integrating factor \( \mu \) must not be zero. If \( \mu(x, y) \equiv 0 \), multiplying any equation by zero gives the trivial equation 0=0, which is not useful for finding solutions to the original equations. Therefore, the suitability of \( \mu \) as an integrating factor ensures the equivalence of the solutions.
05
Conclusion
The equations \( M\,dx + N\,dy = 0 \) and \( \mu M\,dx + \mu N\,dy = 0 \) are equivalent in the sense that a solution to one is a solution to the other, provided that \( \mu eq 0 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Exact Differential Equations
In mathematics, exact differential equations are equations that can be expressed in a specific form that allows the solution process to be relatively straightforward. An exact differential equation takes the form \( M\,dx + N\,dy = 0 \). For an equation to be considered exact, there must exist a potential function \( f(x, y) \) such that the differential of \( f \) is exactly this equation.
This means that \( \frac{\partial f}{\partial x} = M \) and \( \frac{\partial f}{\partial y} = N \). If these conditions hold, the differential equation can be directly integrated, as the potential function acts much like an anti-derivative does in basic calculus.
Being able to determine exactness allows us to find closed-form solutions efficiently, but not all differential equations meet this criterion, leading us to our next concept.
This means that \( \frac{\partial f}{\partial x} = M \) and \( \frac{\partial f}{\partial y} = N \). If these conditions hold, the differential equation can be directly integrated, as the potential function acts much like an anti-derivative does in basic calculus.
Being able to determine exactness allows us to find closed-form solutions efficiently, but not all differential equations meet this criterion, leading us to our next concept.
Non-exact Differential Equations
Non-exact differential equations are those where the conditions for exactness are not satisfied. This means that there are no potential functions \( f(x, y) \) for which \( M\,dx + N\,dy = 0 \) directly corresponds to \( df \). These equations cannot be solved as easily as exact equations because they cannot be integrated directly using simple calculus concepts.
Fortunately, we have a tool called an "integrating factor" which can transform a non-exact equation into an exact one. The integrating factor is a function, often denoted by \( \mu(x, y) \), that when multiplied with the non-exact equation, results in an equation that becomes exact.
The process of finding an integrating factor is sometimes complex, but once found, it greatly simplifies solving the differential equation.
Fortunately, we have a tool called an "integrating factor" which can transform a non-exact equation into an exact one. The integrating factor is a function, often denoted by \( \mu(x, y) \), that when multiplied with the non-exact equation, results in an equation that becomes exact.
The process of finding an integrating factor is sometimes complex, but once found, it greatly simplifies solving the differential equation.
Solutions of Differential Equations
The solution of a differential equation is a function or a set of functions that satisfy the equation. For instance, in our context, if \( M\,dx + N\,dy = 0 \) is exact, there exists a potential function \( f(x, y) \) such that \( df = 0 \). Solving this leads us to \( f(x, y) = C \), where \( C \) is an integration constant. This expression represents a family of solutions that define curves on a plane.
Using the integrating factor technique for non-exact differential equations involves identifying the factor \( \mu(x, y) \) that can transform the equation into an exact form. This allows us to apply the same solution process as for exact differential equations. Hence, an integrating factor expands the solvability of differential equations significantly.
It is crucial that the integrating factor never equals zero, as this would render the equation unsolvable in the traditional sense.
Using the integrating factor technique for non-exact differential equations involves identifying the factor \( \mu(x, y) \) that can transform the equation into an exact form. This allows us to apply the same solution process as for exact differential equations. Hence, an integrating factor expands the solvability of differential equations significantly.
It is crucial that the integrating factor never equals zero, as this would render the equation unsolvable in the traditional sense.
Potential Function
A potential function is integral to solving exact differential equations. It is a function \( f(x, y) \) whose existence guarantees that a differential equation is exact. When dealing with an exact equation, \( M\,dx + N\,dy = 0 \), the potential function \( f \) fulfills \( df = M\,dx + N\,dy \).
Finding the potential function involves partial integration of the components \( M \) and \( N \). This process links calculus to differential equations, specifically showcasing how we can solve them when they represent the derivative of some function.
Understanding potential functions offers a conceptual insight into how changing the perspective on a differential equation鈥攆rom viewing it as a relation between derivatives to viewing it as a differential of a single function鈥攃an drastically simplify solving it.
Finding the potential function involves partial integration of the components \( M \) and \( N \). This process links calculus to differential equations, specifically showcasing how we can solve them when they represent the derivative of some function.
Understanding potential functions offers a conceptual insight into how changing the perspective on a differential equation鈥攆rom viewing it as a relation between derivatives to viewing it as a differential of a single function鈥攃an drastically simplify solving it.
