- Determine whether each of the equations is exact. If it is exact, find the solution. (a) \(\displaystyle{(2xy^2+2y) \, dx + (2x^2y+2x) \, dy = 0}\) View Solution (b) \(\displaystyle{\left(1+\ln x + \frac{y}{x}\right) -(1-\ln x)\, y'=0}\) View Solution (c) \(\displaystyle{ (\sin y - y \sin x)\, dx + (\cos x + x \cos y - y) \, dy = 0 }\) View Solution (d) \(\displaystyle{ (e^x \sin y - 2 y \sin x) + (e^x\cos y + 2 \cos x)\, \frac{dy}{dx} = 0 }\) View Solution
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Find the value \(k\) so that the following equation is exact, then find the general solution.
View Solution
\(\displaystyle{ (5x+ky)\,dx + (4x-3y)\, dy = 0 }\)
- Show that \(\mu(x,y)=xy^2\) is an integrating factor for \((2y-6x)\,dx+(3x-4x^2y^{-1})\, dy =0\), and use the integrating factor to solve the diff eqn. View Solution
- Find an Integrating Factor (to make the equation exact), then use it to solve the differential equation. (a) \(\displaystyle{(3x^2y+2xy+y^3) \, dx + (x^2+y^2) \, dy = 0}\) View Solution (b) \(\displaystyle{y \, dx + (2xy-e^{-2y}) \, dy = 0}\) View Solution (c) \(\displaystyle{e^{x+2y} \, dx + \left(2+\frac{2}{y}\right)e^{x+2y} \, dy = 0}\) View Solution (d) \(\displaystyle{(2xy^3-2x^3y^3-4xy^2+2x)\, dx + (3x^2y^2+4y)\, dy =0}\) View Solution