Exact Equations
Additional Examples
- We write the equation in the standard form,
M dx + N dy = 0 . $$ (30xy - 10x - 9y + 4) dx + (15x^2 - 9x + 6y + 11) dy = 0 $$ - We test for exactness. $$\frac{\partial}{\partial y}\left(30xy - 10x - 9y + 4\right) = 30x - 9 = \frac{\partial}{\partial x}\left(15x^2 - 9x + 6y + 11\right) $$ so the equation is exact.
- Write the partial differential equations. $$ \begin{align} \frac{\partial F}{\partial x} &= 30xy - 10x - 9y + 4\\ \frac{\partial F}{\partial y} &= 15x^2 - 9x + 6y + 11 \end{align}$$
- Integrate the first partial differential equation. $$ F(x,y) = \int (30xy - 10x - 9y + 4)\,\partial x = 15x^2y - 5x^2 - 9xy + 4x + C(y) $$
- Integrate the second partial differential equation. $$ F(x,y) = \int (15x^2 - 9x + 6y + 11)\,\partial y = 15x^2y - 9xy + 3y^2 + 11y + \tilde{C}(x) $$
- Equate the expressions for F(x,y). Matching the expressions up, we find $C(y) = 3y^2 + 11y$ and $ \tilde{C}(x) = -5x^2 + 4x. $ So $$ F(x,y) = 15x^2y - 9xy - 5x^2 + 3y^2 + 4x + 11y. $$
- The solution is $F(x,y) = K.$ $$ 15x^2y - 9xy - 5x^2 + 3y^2 + 4x + 11y = K $$
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