Solving Exact Differential Equations: (1 + e^(x/y))dx + e^(x/y)(1 - x/y)dy = 0: 17 flashcards
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1. What is the standard form of a first-order differential equation that may be exact?
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$M\,dx + N\,dy = 0$
2. State the exactness test for $M\,dx + N\,dy = 0$.
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The equation is exact if and only if $\dfrac{\partial M}{\partial y} = \dfrac{\partial N}{\partial x}$.
3. For the equation $(1 + e^{x/y})dx + e^{x/y}(1 - x/y)dy = 0$, what are $M$ and $N$?
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$M = 1 + e^{x/y}$ and $N = e^{x/y}\left(1 - \dfrac{x}{y}\right)$.
4. Compute $\dfrac{\partial M}{\partial y}$ for $M = 1 + e^{x/y}$.
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$\dfrac{\partial M}{\partial y} = -\dfrac{x}{y^2}e^{x/y}$
5. Compute $\dfrac{\partial N}{\partial x}$ for $N = e^{x/y}\left(1 - \dfrac{x}{y}\right)$.
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$\dfrac{\partial N}{\partial x} = -\dfrac{x}{y^2}e^{x/y}$
6. If $M\,dx + N\,dy = 0$ is exact, what function exists and what is the solution?
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There exists $F(x,y)$ with $\partial F/\partial x = M$ and $\partial F/\partial y = N$; the solution is $F(x,y) = C$.
7. How do you begin finding $F(x,y)$ for an exact equation?
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Integrate $M$ with respect to $x$: $F = \int M\,dx + f(y)$.
8. Evaluate $\int (1 + e^{x/y})\,dx$.
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$x + y\,e^{x/y} + f(y)$
9. After integrating $M$ with respect to $x$, how do you determine $f(y)$?
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Differentiate $F$ with respect to $y$, set it equal to $N$, and solve for $f'(y)$.
10. In this problem, what is $f'(y)$ and what does that imply?
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$f'(y) = 0$, so $f(y)$ is a constant that is absorbed into $C$.
11. What is the final implicit solution of $(1 + e^{x/y})dx + e^{x/y}(1 - x/y)dy = 0$?
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$x + y\,e^{x/y} = C$
12. How do you verify that $F(x,y) = x + y\,e^{x/y}$ is correct?
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Check $\partial F/\partial x = M$ and $\partial F/\partial y = N$.
13. Why is the solution $x + y\,e^{x/y} = C$ called implicit?
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Because $y$ is not isolated; the relationship between $x$ and $y$ is given as an equation.
14. What does the exactness condition $\partial M/\partial y = \partial N/\partial x$ correspond to in physics?
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It is the same test for whether a vector field is conservative (a conservative force field).
15. In thermodynamics, what do exact differentials correspond to?
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State functions such as internal energy and entropy.
16. What is the role of the constant of integration when finding $F(x,y)$?
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An arbitrary function $f(y)$ appears after integrating $M$ with respect to $x$; it is determined up to a constant that merges into $C$.
17. If the exactness test fails, what would be the next step?
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The equation is not exact and would require an integrating factor.
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