Exercises

(1)
Which equations, explicitly as written, are NOT in an exact form? Select all that apply.
(a)
\((x+3y)\,\mathrm {d}x+(3x-y)\,\mathrm {d}y=0\).
(b)
\((x-2xy)\,\mathrm {d}x+(y-x^2)\,\mathrm {d}y=0\).
(c)
\((4x^3+3x^2y)\,\mathrm {d}x+(x^3+\cos {(y)})\,\mathrm {d}y=0\).
(d)
\(ye^{xy}\,\mathrm {d}x+\left (\ln {(y)}+xe^{xy}\right )\,\mathrm {d}y=0\).
(e)
\(ye^{xy}\,\mathrm {d}x+\left (\ln {(x)}+xe^{xy}\right )\,\mathrm {d}y=0\).
(f)
\((x^2-2xy)\,\mathrm {d}x+(x^2+\sin {(y)})\,\mathrm {d}y=0\).
(2)
For what value of \(a\), to the nearest tenth, is the following differential equation is in an exact form, explicitly as written?
\[ \left (2axy^3+y\right )\,\mathrm {d}x+\left (15x^2y^2+x-\frac {1}{y}\right )\,\mathrm {d}y=0.\]
(3)
Suppose \(y(x)\) satisfies the IVP \(\displaystyle \frac {y}{x}\,\mathrm {d}x +\left (\ln {(x)}-\frac {1}{y-1}-2y\right )\, \mathrm {d}y=0\), \(y(1)=2\). Find the solution to \(y(x)=3\), to the nearest hundredth.
(4)
If \(M\,\mathrm {d}x+N\,\mathrm {d}y=0\) is not exact, but \(p=(M_{y}-N_{x})/N\) is independent of \(y\) then \(\displaystyle \mu (x)= e^{P(x)}\), where \(P'(x)=p(x)\), is an integrating factor of \(M\,\mathrm {d}x+N\,\mathrm {d}y=0\) (meaning that the equation becomes exact when both sides are multiplied by \(\mu \)).

Find the integrating factor \(\mu (x)\) for \((4xy+2y+5)\,\mathrm {d}x+ 2x^2\,\mathrm {d}y=0\) such that \(\mu (-1)=1\). Find \(\mu (2)\) to the nearest hundredth.

(5)
A specialized piston-cylinder device contains a gas where the pressure and volume are governed by
\[ \left [ P+ \dfrac {P}{V}e^{\dfrac {P}{V}} \right ] \mathrm {d}V + \left [ V\ln (V) - e^{\dfrac {P}{V}} \right ] \mathrm {d}P = 0. \]

Determine the partial derivative \(P_V\).

Hint: using implicit differentiation, if \(F(P,V)=C\) then \(P_V=-\dfrac {F_V}{F_P}\).

(a)
\(\dfrac {PV+Pe^{P/V}}{V^2\ln (V)-Ve^{P/V}}\)
(b)
\(-\dfrac {PV+Pe^{P/V}}{V^2\ln (V)-Ve^{P/V}}\)
(c)
\(\dfrac {V^2\ln (V)-Ve^{P/V}}{PV+Pe^{P/V}}\)
(d)
\(-\dfrac {V^2\ln (V)-Ve^{P/V}}{PV+Pe^{P/V}}\)
(e)
\(\dfrac {PV+Pe^{P/V}}{V^2\ln (V)+Ve^{P/V}}\)
(f)
\(-\dfrac {PV+Pe^{P/V}}{V^2\ln (V)+Ve^{P/V}}\)
(g)
\(\dfrac {V^2\ln (V)+Ve^{P/V}}{PV+Pe^{P/V}}\)
(h)
\(-\dfrac {V^2\ln (V)+Ve^{P/V}}{PV+Pe^{P/V}}\)