Determine whether the following differential form is exact or not. If it is not exact then find an integrating factor \(\rho\) that transforms it into an exact differential form.
\(2xy \, \dx + (x^2 - e^y) \, \dy = 0\)
\((xy^2 + 3x^2y) \, \dx + (x+y)x^2 \, \dy = 0\)
\((2xy + 2x^2y + y^2) \, \dx + (x^2 + y) \, \dy = 0\)
\(\frac{\dy}{\dx} = \frac{-y}{(2x - ye^y)}\)
\((2x + 3) + 2(y - 2) \frac{\dy}{\dx} = 0\)
\(\frac{\dy}{\dx} = \frac{-(2xy^2 + 2y)}{(2x^2y + 2x)}\)
Find an implicit general solution to the following differential equations.
(Make sure to check whether the differential form is exact or not, first!)
\((3x^2 - 2xy + 2) \, \dx + (6y^2 - x^2 + 3) \, \dy = 0\)
\(\left(\dfrac{y}{x} + 6x\right) \, \dx + (\log(x) - 2) \, \dy = 0 \,, \qquad x > 0\)
\((x + 2y) \, \dx + (2x - y) \, \dy = 0\)
\((3xy-y^2)\,\dx + (x^2 - xy)\,\dy = 0\)
\((y^2 - xy)\,\dx + (3xy - x^2)\,\dy = 0\)
\(-(2x+y)\,\dx = (x + 9y)\,\dy\)
\(\left( \dfrac{x^3}{y^2} + \dfrac{2}{y} \right) \, \dx + \left( \dfrac{2x}{y^2} + y \right) \, \dy = 0\)
\((3x-2y)\,\dx + (y - 2x)\,\dy = 0\)
\(\dfrac{\dy}{\dx} = e^{2x} + y - 2\)
\((y e^{2xy})\,\dx + (x e^{2xy} + y) \, \dy = 0\)
\((e^{y}\cot(x) + 2x\csc(x))\,\dx + e^{y}\,\dy = 0\)
For #18-20, solve the initial-value problems.
(You may leave your answers in implicit form)
\((3x-2y)\,\dx + (y - 2x)\,\dy = 0 \qquad y(1) = 3 \,.\)
\((2xy-9x^2)\,\dx + (2y+x^2+1) \, \dy = 0\)
\(y e^{2xy} \, \dx + (x e^{2xy} + y) \, \dy = 0 \,, \qquad y(1) = 0 \,.\)
\((x + 2y)\,\dx + (2x-y)\,\dy = 0 \,, \qquad y(2) = 2 \,.\)
Show that a separated differential form (i.e. \(\dfrac{1}{g(y)} \, \dy = f(x) \, \dx\)) is exact. Assume \(g(y) \neq 0\) for every \(y\). Find an implicit general solution using \(f(x)\) and \(g(y)\).
(Note: Let \(F(x)\) denote a primitive of \(f(x)\) and \(G(y)\) denote a primitive of \(\dfrac{1}{g(y)}\).)
Consider a differential form \(M(x,y)\,\dx+N(x,y)\,\dy=0\) such that \(\dfrac{\partial_y M(x,y) - \partial_x N(x,y)}{N(x,y)}\) is a function of only \(x\). Show that there exists an integrating factor \(\rho(x)\) that satisfies \[\rho' = \frac{\partial_y M(x,y) - \partial_x N(x,y)}{N(x,y)} \, \rho \,.\]
Consider a differential form \(M(x,y)\,\dx+N(x,y)\,\dy=0\) such that \(\dfrac{\partial_x N(x,y) - \partial_y M(x,y)}{M(x,y)}\) is a function of only \(y\). Show that there exists an integrating factor \(\rho(y)\) that satisfies \[\rho' = \frac{\partial_x N(x,y) - \partial_y M(x,y)}{M(x,y)} \, \rho \,.\]
What value of \(\alpha\) makes this exact? \((\alpha\,x^3 +\alpha\,y^2x)\,\dx + (4yx^2+y^4) \, \dy = 0\)
Solve the previous problem using the value of \(\alpha\) that makes the differential form exact.
What value of \(\alpha\),\(\beta\) make this exact? \((\frac{\alpha\,y}{x}+\beta\,x)\,\dx + (log(\beta\,x)+6) \, \dy = 0\)