Kansas State University

Mathematics Department

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Exact Equations

Additional Examples

Solve the following initial value problem, $$\begin{align} \frac{dy}{dx} &= \frac{-15y^2 + 20y - 1}{30xy - 20x - 6y - 1} \\ y(0) &= 0 \end{align}$$ 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. $$ (15y^2 - 20y + 1) dx + (30xy - 20x - 6y - 1) dy = 0 $$
  2. We test for exactness. $$\frac{\partial}{\partial y}\left(15y^2 - 20y + 1\right) = 30y - 20 = \frac{\partial}{\partial x}\left(30xy - 20x - 6y - 1\right) $$ so the equation is exact.

  3. Write the partial differential equations. $$ \begin{align} \frac{\partial F}{\partial x} &= 15y^2 - 20y + 1\\ \frac{\partial F}{\partial y} &= 30xy - 20x - 6y - 1 \end{align}$$
  4. Integrate the first partial differential equation. $$ F(x,y) = \int (15y^2 - 20y + 1)\,\partial x = 15xy^2 - 20xy + x + C(y) $$
  5. Integrate the second partial differential equation. $$ F(x,y) = \int (30xy - 20x - 6y - 1)\,\partial y = 15xy^2 - 20xy - 3y^2 - y + \tilde{C}(x) $$
  6. Equate the expressions for F(x,y).

    Matching the expressions up, we find $C(y) = -3y^2 - y$ and $ \tilde{C}(x) = x. $ So $$ F(x,y) = 15xy^2 - 20xy - 3y^2 + x - y. $$

  7. The solution is $F(x,y) = K.$ $$ 15xy^2 - 20xy - 3y^2 + x - y = K $$
Now we plug in the initial values $x = 0$ and $y = 0$ and solve for $K = 0$. So the solution to the initial value problem is $$ 15xy^2 - 20xy - 3y^2 + x - y = 0 $$ You may reload this page to generate additional examples.

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