Kansas State University

Mathematics Department

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Exact Equations

Additional Examples

Solve the following initial value problem, $$\begin{align} \frac{dy}{dx} &= \frac{-20xy - 10x + 2y + 6}{15y^2 + 10x^2 - 2y - 2x - 5} \\ y(0) &= 1 \end{align}$$ 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. $$ (20xy + 10x - 2y - 6) dx + (15y^2 + 10x^2 - 2y - 2x - 5) dy = 0 $$
  2. We test for exactness. $$\frac{\partial}{\partial y}\left(20xy + 10x - 2y - 6\right) = 20x - 2 = \frac{\partial}{\partial x}\left(15y^2 + 10x^2 - 2y - 2x - 5\right) $$ so the equation is exact.

  3. Write the partial differential equations. $$ \begin{align} \frac{\partial F}{\partial x} &= 20xy + 10x - 2y - 6\\ \frac{\partial F}{\partial y} &= 15y^2 + 10x^2 - 2y - 2x - 5 \end{align}$$
  4. Integrate the first partial differential equation. $$ F(x,y) = \int (20xy + 10x - 2y - 6)\,\partial x = 10x^2y + 5x^2 - 2xy - 6x + C(y) $$
  5. Integrate the second partial differential equation. $$ F(x,y) = \int (15y^2 + 10x^2 - 2y - 2x - 5)\,\partial y = 5y^3 + 10x^2y - y^2 - 2xy - 5y + \tilde{C}(x) $$
  6. Equate the expressions for F(x,y).

    Matching the expressions up, we find $C(y) = 5y^3 - y^2 - 5y$ and $ \tilde{C}(x) = 5x^2 - 6x. $ So $$ F(x,y) = 5y^3 + 10x^2y - 2xy + 5x^2 - y^2 - 6x - 5y. $$

  7. The solution is $F(x,y) = K.$ $$ 5y^3 + 10x^2y - 2xy + 5x^2 - y^2 - 6x - 5y = K $$
Now we plug in the initial values $x = 0$ and $y = 1$ and solve for $K = -1$. So the solution to the initial value problem is $$ 5y^3 + 10x^2y - 2xy + 5x^2 - y^2 - 6x - 5y = -1 $$ You may reload this page to generate additional examples.

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