Kansas State University

Mathematics Department

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Exact Equations

Additional Examples

Solve the following initial value problem, $$\begin{align} \frac{dy}{dx} &= \frac{-6xy - 10x - 15y - 30}{3x^2 + 15x + 8y + 10} \\ y(2) &= 0 \end{align}$$ 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. $$ (6xy + 10x + 15y + 30) dx + (3x^2 + 15x + 8y + 10) dy = 0 $$
  2. We test for exactness. $$\frac{\partial}{\partial y}\left(6xy + 10x + 15y + 30\right) = 6x + 15 = \frac{\partial}{\partial x}\left(3x^2 + 15x + 8y + 10\right) $$ so the equation is exact.

  3. Write the partial differential equations. $$ \begin{align} \frac{\partial F}{\partial x} &= 6xy + 10x + 15y + 30\\ \frac{\partial F}{\partial y} &= 3x^2 + 15x + 8y + 10 \end{align}$$
  4. Integrate the first partial differential equation. $$ F(x,y) = \int (6xy + 10x + 15y + 30)\,\partial x = 3x^2y + 5x^2 + 15xy + 30x + C(y) $$
  5. Integrate the second partial differential equation. $$ F(x,y) = \int (3x^2 + 15x + 8y + 10)\,\partial y = 3x^2y + 15xy + 4y^2 + 10y + \tilde{C}(x) $$
  6. Equate the expressions for F(x,y).

    Matching the expressions up, we find $C(y) = 4y^2 + 10y$ and $ \tilde{C}(x) = 5x^2 + 30x. $ So $$ F(x,y) = 3x^2y + 15xy + 5x^2 + 4y^2 + 30x + 10y. $$

  7. The solution is $F(x,y) = K.$ $$ 3x^2y + 15xy + 5x^2 + 4y^2 + 30x + 10y = K $$
Now we plug in the initial values $x = 2$ and $y = 0$ and solve for $K = 80$. So the solution to the initial value problem is $$ 3x^2y + 15xy + 5x^2 + 4y^2 + 30x + 10y = 80 $$ You may reload this page to generate additional examples.

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