Kansas State University

Mathematics Department

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Exact Equations

Additional Examples

Solve the following initial value problem, $$\begin{align} \frac{dy}{dx} &= \frac{40xy - 28x + 4y - 4}{12y^2 - 20x^2 - 2y - 4x - 13} \\ y(-1) &= -4 \end{align}$$ This can be written as an exact equation. First we find the general solution following the paradigm.

  1. We write the equation in the standard form, M dx + N dy = 0. $$ (-40xy + 28x - 4y + 4) dx + (12y^2 - 20x^2 - 2y - 4x - 13) dy = 0 $$
  2. We test for exactness. $$\frac{\partial}{\partial y}\left(-40xy + 28x - 4y + 4\right) = -40x - 4 = \frac{\partial}{\partial x}\left(12y^2 - 20x^2 - 2y - 4x - 13\right) $$ so the equation is exact.

  3. Write the partial differential equations. $$ \begin{align} \frac{\partial F}{\partial x} &= -40xy + 28x - 4y + 4\\ \frac{\partial F}{\partial y} &= 12y^2 - 20x^2 - 2y - 4x - 13 \end{align}$$
  4. Integrate the first partial differential equation. $$ F(x,y) = \int (-40xy + 28x - 4y + 4)\,\partial x = -20x^2y + 14x^2 - 4xy + 4x + C(y) $$
  5. Integrate the second partial differential equation. $$ F(x,y) = \int (12y^2 - 20x^2 - 2y - 4x - 13)\,\partial y = 4y^3 - 20x^2y - y^2 - 4xy - 13y + \tilde{C}(x) $$
  6. Equate the expressions for F(x,y).

    Matching the expressions up, we find $C(y) = 4y^3 - y^2 - 13y$ and $ \tilde{C}(x) = 14x^2 + 4x. $ So $$ F(x,y) = 4y^3 - 20x^2y - 4xy + 14x^2 - y^2 + 4x - 13y. $$

  7. The solution is $F(x,y) = K.$ $$ 4y^3 - 20x^2y - 4xy + 14x^2 - y^2 + 4x - 13y = K $$
Now we plug in the initial values $x = -1$ and $y = -4$ and solve for $K = -146$. So the solution to the initial value problem is $$ 4y^3 - 20x^2y - 4xy + 14x^2 - y^2 + 4x - 13y = -146 $$ You may reload this page to generate additional examples.

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