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Exact Equations

Additional Examples

Solve the following initial value problem, $$\begin{align} \frac{dy}{dx} &= \frac{10xy + 4x - 3y}{-5x^2 + 3x - 7} \\ y(-2) &= 1 \end{align}$$ 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. $$ (-10xy - 4x + 3y) dx + (-5x^2 + 3x - 7) dy = 0 $$
  2. We test for exactness. $$\frac{\partial}{\partial y}\left(-10xy - 4x + 3y\right) = -10x + 3 = \frac{\partial}{\partial x}\left(-5x^2 + 3x - 7\right) $$ so the equation is exact.

  3. Write the partial differential equations. $$ \begin{align} \frac{\partial F}{\partial x} &= -10xy - 4x + 3y\\ \frac{\partial F}{\partial y} &= -5x^2 + 3x - 7 \end{align}$$
  4. Integrate the first partial differential equation. $$ F(x,y) = \int (-10xy - 4x + 3y)\,\partial x = -5x^2y - 2x^2 + 3xy + C(y) $$
  5. Integrate the second partial differential equation. $$ F(x,y) = \int (-5x^2 + 3x - 7)\,\partial y = -5x^2y + 3xy - 7y + \tilde{C}(x) $$
  6. Equate the expressions for F(x,y).

    Matching the expressions up, we find $C(y) = -7y$ and $ \tilde{C}(x) = -2x^2. $ So $$ F(x,y) = -5x^2y - 2x^2 + 3xy - 7y. $$

  7. The solution is $F(x,y) = K.$ $$ -5x^2y - 2x^2 + 3xy - 7y = K $$
Now we plug in the initial values $x = -2$ and $y = 1$ and solve for $K = -41$. So the solution to the initial value problem is $$ -5x^2y - 2x^2 + 3xy - 7y = -41 $$ You may reload this page to generate additional examples.


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