Section 2.6 – Exact Equations and Integrating Factors
Exercises
Directions: You should try to solve each problem first, and then click "Reveal Answer" to check your answer. You can click "Watch Video" if you need help with a problem.
1. Determine if the equations are exact and solve the ones that are:
- \((2x+5y) dx+(5x-6y) dy=0\)
- \(1+\dfrac{y}{x}-\dfrac{1}{x}y'=0\)
- \((\sin(2t)+2y) dy+(2y\cos(2t)-6t^2) dt=0\)
- Equation is exact. The solution is \(\quad x^2+5xy-3y^2=C.\)
- Equation is not exact.
- Equation is exact. The solution is \(\quad y\sin(2t)+y^2-2t^3=C.\)
2. Show that the following equations are not exact. However, if you multiply by the given integrating factor, the resulting equation is exact.
- \((x^2+y^2-x) dx-ydy=0\qquad\) \(\mu(x,y)=\dfrac{1}{x^2+y^2}\)
- \(3(y+1)dx-2xdy=0,\qquad\) \(\mu(x,y)=\dfrac{y+1}{x^4}\)
For verification, see the video.
3. Find an integrating factor for the equation \[(3xy+y^2)+(x^2+xy)y'=0\] and then solve the equation.
\(\mu=x, \quad\) \(x^3y+\dfrac{1}{2}x^2y^2=C\)
4. Solve the IVP \((3x^2+2xy^2)dx+2x^2ydy=0,\quad\) \(y(2)=-3.\)
\(y=-\sqrt{\dfrac{44-x^3}{x^2}}\)
6. Solve \(y'=\displaystyle\frac{(1+x)e^x}{xe^x-ye^y}.\)
\(\dfrac{y^2}{2}+xe^xe^{-y}=C\)
7. Solve \[ydx = (y^2 + x^2 + x)dy,\] if we know there is an integrating factor of the form \(\mu=\phi(x^2+y^2).\)
\(\arctan\left(\dfrac{x}{y}\right)-y=C\)