Exact Equations
Additional Examples
- We write the equation in the standard form,
M dx + N dy = 0 . $$ (32xy + 48x - 8y - 5) dx + (16x^2 - 8x - 2y - 5) dy = 0 $$ - We test for exactness. $$\frac{\partial}{\partial y}\left(32xy + 48x - 8y - 5\right) = 32x - 8 = \frac{\partial}{\partial x}\left(16x^2 - 8x - 2y - 5\right) $$ so the equation is exact.
- Write the partial differential equations. $$ \begin{align} \frac{\partial F}{\partial x} &= 32xy + 48x - 8y - 5\\ \frac{\partial F}{\partial y} &= 16x^2 - 8x - 2y - 5 \end{align}$$
- Integrate the first partial differential equation. $$ F(x,y) = \int (32xy + 48x - 8y - 5)\,\partial x = 16x^2y + 24x^2 - 8xy - 5x + C(y) $$
- Integrate the second partial differential equation. $$ F(x,y) = \int (16x^2 - 8x - 2y - 5)\,\partial y = 16x^2y - 8xy - y^2 - 5y + \tilde{C}(x) $$
- Equate the expressions for F(x,y). Matching the expressions up, we find $C(y) = -y^2 - 5y$ and $ \tilde{C}(x) = 24x^2 - 5x. $ So $$ F(x,y) = 16x^2y - 8xy + 24x^2 - y^2 - 5x - 5y. $$
- The solution is $F(x,y) = K.$ $$ 16x^2y - 8xy + 24x^2 - y^2 - 5x - 5y = K $$
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