Kansas State University

Mathematics Department

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Exact Equations

Additional Examples

Solve the following initial value problem, $$\begin{align} \frac{dy}{dx} &= \frac{10y^2 - 2y - 6x + 6}{-20xy + 2x + 30y - 9} \\ y(0) &= -3 \end{align}$$ 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. $$ (-10y^2 + 2y + 6x - 6) dx + (-20xy + 2x + 30y - 9) dy = 0 $$
  2. We test for exactness. $$\frac{\partial}{\partial y}\left(-10y^2 + 2y + 6x - 6\right) = -20y + 2 = \frac{\partial}{\partial x}\left(-20xy + 2x + 30y - 9\right) $$ so the equation is exact.

  3. Write the partial differential equations. $$ \begin{align} \frac{\partial F}{\partial x} &= -10y^2 + 2y + 6x - 6\\ \frac{\partial F}{\partial y} &= -20xy + 2x + 30y - 9 \end{align}$$
  4. Integrate the first partial differential equation. $$ F(x,y) = \int (-10y^2 + 2y + 6x - 6)\,\partial x = -10xy^2 + 2xy + 3x^2 - 6x + C(y) $$
  5. Integrate the second partial differential equation. $$ F(x,y) = \int (-20xy + 2x + 30y - 9)\,\partial y = -10xy^2 + 2xy + 15y^2 - 9y + \tilde{C}(x) $$
  6. Equate the expressions for F(x,y).

    Matching the expressions up, we find $C(y) = 15y^2 - 9y$ and $ \tilde{C}(x) = 3x^2 - 6x. $ So $$ F(x,y) = -10xy^2 + 2xy + 3x^2 + 15y^2 - 6x - 9y. $$

  7. The solution is $F(x,y) = K.$ $$ -10xy^2 + 2xy + 3x^2 + 15y^2 - 6x - 9y = K $$
Now we plug in the initial values $x = 0$ and $y = -3$ and solve for $K = 162$. So the solution to the initial value problem is $$ -10xy^2 + 2xy + 3x^2 + 15y^2 - 6x - 9y = 162 $$ You may reload this page to generate additional examples.

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