Kansas State University

Mathematics Department

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Exact Equations

Additional Examples

Solve the following initial value problem, $$\begin{align} \frac{dy}{dx} &= \frac{-9y^2 + 15y - 6x + 4}{-12y^2 + 18xy + 24y - 15x - 17} \\ y(-1) &= -3 \end{align}$$ 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. $$ (9y^2 - 15y + 6x - 4) dx + (-12y^2 + 18xy + 24y - 15x - 17) dy = 0 $$
  2. We test for exactness. $$\frac{\partial}{\partial y}\left(9y^2 - 15y + 6x - 4\right) = 18y - 15 = \frac{\partial}{\partial x}\left(-12y^2 + 18xy + 24y - 15x - 17\right) $$ so the equation is exact.

  3. Write the partial differential equations. $$ \begin{align} \frac{\partial F}{\partial x} &= 9y^2 - 15y + 6x - 4\\ \frac{\partial F}{\partial y} &= -12y^2 + 18xy + 24y - 15x - 17 \end{align}$$
  4. Integrate the first partial differential equation. $$ F(x,y) = \int (9y^2 - 15y + 6x - 4)\,\partial x = 9xy^2 - 15xy + 3x^2 - 4x + C(y) $$
  5. Integrate the second partial differential equation. $$ F(x,y) = \int (-12y^2 + 18xy + 24y - 15x - 17)\,\partial y = -4y^3 + 9xy^2 + 12y^2 - 15xy - 17y + \tilde{C}(x) $$
  6. Equate the expressions for F(x,y).

    Matching the expressions up, we find $C(y) = -4y^3 + 12y^2 - 17y$ and $ \tilde{C}(x) = 3x^2 - 4x. $ So $$ F(x,y) = -4y^3 + 9xy^2 - 15xy + 3x^2 + 12y^2 - 4x - 17y. $$

  7. The solution is $F(x,y) = K.$ $$ -4y^3 + 9xy^2 - 15xy + 3x^2 + 12y^2 - 4x - 17y = K $$
Now we plug in the initial values $x = -1$ and $y = -3$ and solve for $K = 148$. So the solution to the initial value problem is $$ -4y^3 + 9xy^2 - 15xy + 3x^2 + 12y^2 - 4x - 17y = 148 $$ You may reload this page to generate additional examples.

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