Kansas State University

Mathematics Department

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Exact Equations

Additional Examples

Solve the following initial value problem, $$\begin{align} \frac{dy}{dx} &= \frac{-16y^2 - 4y - 8x + 8}{6y^2 + 32xy - 18y + 4x - 6} \\ y(2) &= 4 \end{align}$$ 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. $$ (16y^2 + 4y + 8x - 8) dx + (6y^2 + 32xy - 18y + 4x - 6) dy = 0 $$
  2. We test for exactness. $$\frac{\partial}{\partial y}\left(16y^2 + 4y + 8x - 8\right) = 32y + 4 = \frac{\partial}{\partial x}\left(6y^2 + 32xy - 18y + 4x - 6\right) $$ so the equation is exact.

  3. Write the partial differential equations. $$ \begin{align} \frac{\partial F}{\partial x} &= 16y^2 + 4y + 8x - 8\\ \frac{\partial F}{\partial y} &= 6y^2 + 32xy - 18y + 4x - 6 \end{align}$$
  4. Integrate the first partial differential equation. $$ F(x,y) = \int (16y^2 + 4y + 8x - 8)\,\partial x = 16xy^2 + 4xy + 4x^2 - 8x + C(y) $$
  5. Integrate the second partial differential equation. $$ F(x,y) = \int (6y^2 + 32xy - 18y + 4x - 6)\,\partial y = 2y^3 + 16xy^2 - 9y^2 + 4xy - 6y + \tilde{C}(x) $$
  6. Equate the expressions for F(x,y).

    Matching the expressions up, we find $C(y) = 2y^3 - 9y^2 - 6y$ and $ \tilde{C}(x) = 4x^2 - 8x. $ So $$ F(x,y) = 2y^3 + 16xy^2 + 4xy + 4x^2 - 9y^2 - 8x - 6y. $$

  7. The solution is $F(x,y) = K.$ $$ 2y^3 + 16xy^2 + 4xy + 4x^2 - 9y^2 - 8x - 6y = K $$
Now we plug in the initial values $x = 2$ and $y = 4$ and solve for $K = 504$. So the solution to the initial value problem is $$ 2y^3 + 16xy^2 + 4xy + 4x^2 - 9y^2 - 8x - 6y = 504 $$ You may reload this page to generate additional examples.

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