| # | original ode | \(y=xf\left ( p\right ) +g\left ( p\right ) \) | \(f\left ( p\right ) \) | \(g\left ( p\right ) \) | type |
| \(1\) | \(x\left ( y^{\prime }\right ) ^{2}-yy^{\prime }=-1\) | \(y=xp+\frac {1}{p}\) | \(p\) | \(\frac {1}{p}\) | Clairaut |
| \(2\) | \(y=xy^{\prime }-\left ( y^{\prime }\right ) ^{2}\) | \(y=xp-p^{2}\) | \(p\) | \(-p^{2}\) | Clairaut |
| \(3\) | \(y=xy^{\prime }-\frac {1}{4}\left ( y^{\prime }\right ) ^{2}\) | \(y=xp-\frac {1}{4}p^{2}\) | \(p\) | \(-\frac {1}{4}p^{2}\) | Clairaut |
| \(4\) | \(y=x\left ( y^{\prime }\right ) ^{2}\) | \(y=xp^{2}\) | \(p^{2}\) | \(0\) | d’Alembert |
| \(5\) | \(y=x+\left ( y^{\prime }\right ) ^{2}\) | \(y=x+p^{2}\) | \(1\) | \(p^{2}\) | d’Alembert |
| \(6\) | \(\left ( y^{\prime }\right ) ^{2}-1-x-y=0\) | \(y=-x+\left ( p^{2}-1\right ) \) | \(-1\) | \(\left ( p^{2}-1\right ) \) | d’Alembert |
| \(7\) | \(yy^{\prime }-\left ( y^{\prime }\right ) ^{2}=x\) | \(y=\frac {1}{p}x+p\) | \(\frac {1}{p}\) | \(p\) | d’Alembert |
| \(8\) | \(y=x\left ( y^{\prime }\right ) ^{2}+\left ( y^{\prime }\right ) ^{2}\) | \(y=xp^{2}+p^{2}\) | \(p^{2}\) | \(p^{2}\) | d’Alembert |
| \(9\) | \(y=\frac {x}{a}y^{\prime }+\frac {b}{ay^{\prime }}\) | \(y=\frac {x}{a}p+\frac {b}{a}\frac {1}{p}\) | \(\frac {p}{a}\) | \(\frac {b}{a}\frac {1}{p}\) | d’Alembert |
| \(10\) | \(y=x\left ( y^{\prime }+a\sqrt {1+\left ( y^{\prime }\right ) ^{2}}\right ) \) | \(y=x\left ( p+a\sqrt {1+p^{2}}\right ) \) | \(p+a\sqrt {1+p^{2}}\) | \(0\) | d’Alembert |
| \(11\) | \(y=x+\left ( y^{\prime }\right ) ^{2}\left ( 1-\frac {2}{3}y^{\prime }\right ) \) | \(y=x+p^{2}\left ( 1-\frac {2}{3}p\right ) \) | \(1\) | \(p^{2}\left ( 1-\frac {2}{3}p\right ) \) | d’Alembert |
| \(12\) | \(y=2x-\frac {1}{2}\ln \left ( \frac {\left ( y^{\prime }\right ) ^{2}}{y^{\prime }-1}\right ) \) | \(y=2x-\frac {1}{2}\ln \left ( \frac {p^{2}}{p-1}\right ) \) | \(2\) | \(-\frac {1}{2}\ln \left ( \frac {p^{2}}{p-1}\right ) \) | d’Alembert |
| \(13\) | \(\left ( y^{\prime }\right ) ^{2}-x\left ( y^{\prime }\right ) ^{2}+y\left ( 1+y^{\prime }\right ) -xy^{\prime }=0\) | \(y=\frac {xp+xp^{2}-p^{2}}{p+1}=xp-\frac {p^{2}}{p+1}\) | \(p\) | \(-\frac {p^{2}}{p+1}\) | Clairaut |
| \(14\) | \(x\left ( y^{\prime }\right ) ^{2}+\left ( x-y\right ) y^{\prime }+1-y=0\) | \(y=xp+\frac {1}{1+p}\) | \(p\) | \(\frac {1}{1+p}\) | Clairaut |
| \(15\) | \(xyy^{\prime }=y^{2}+x\sqrt {4x^{2}+y^{2}}\) | \(y=\operatorname {RootOf}\left ( h(p)\right ) x\) | \(\operatorname {RootOf}\left ( h(p)\right ) \) | \(0\) | d’Alembert |
| \(16\) | \(\ln \left ( \cos y^{\prime }\right ) +y^{\prime }\tan y^{\prime }=y\) | \(y=\ln \left ( \cos p\right ) +p\tan p\) | \(0\) | \(\ln \left ( \cos p\right ) +p\tan p\) | d’Alembert |
| \(17\) | \(x\left ( y^{\prime }\right ) ^{2}-2yy^{\prime }+4x=0\) | \(y=x\left ( \frac {1}{2}p+\frac {2}{p}\right ) \) | \(\frac {1}{2}p+\frac {2}{p}\) | \(0\) | d’Alembert |
| \(18\) | \(x-yy^{\prime }=a\left ( y^{\prime }\right ) ^{2}\) | \(y=\frac {x}{p}-ap\) | \(\frac {1}{p}\) | \(-ap\) | d’Alembert |
| \(19\) | \(y=xF\left ( p\right ) +G\left ( p\right ) \) | \(y=xF\left ( p\right ) +G\left ( p\right ) \) | \(F\left ( p\right ) \) | \(G\left ( p\right ) \) | d’Alembert |
| \(20\) | \(y^{\prime }=-\frac {x}{2}-1+\frac {1}{2}\sqrt {x^{2}+4x+4y}\) | \(y=xp+\left ( 1+2p+p^{2}\right ) \) | \(p\) | \(1+2p+p^{2}\) | Clairaut |
| \(21\) | \(\frac {y^{\prime }y}{1+\frac {1}{2}\sqrt {1+\left ( y^{\prime }\right ) ^{2}}}=-x\) | \(y=-x\left ( \frac {2+\sqrt {1+p^{2}}}{2p}\right ) \) | \(-\left ( \frac {2+\sqrt {1+p^{2}}}{2p}\right ) \) | \(0\) | d’Alembert |
| \(22\) | \(x\left ( y^{\prime }\right ) ^{3}=yy^{\prime }+1\) | \(y=xp^{2}-\frac {1}{p}\) | \(p^{2}\) | \(-\frac {1}{p}\) | d’Alembert |
| \(23\) | \(\left ( y^{\prime }\right ) ^{2}-2yy^{\prime }=2x\) | \(y=-x\frac {1}{p}+\frac {1}{2}p\) | \(-\frac {1}{p}\) | \(\frac {1}{2}p\) | d’Alembert |
| \(24\) | \(xy^{\prime }-y=\sqrt {x^{2}-y^{2}}\) | \(y=x\left ( \frac {p}{2}\pm \frac {1}{2}\sqrt {2-p^{2}}\right ) \) | \(\frac {p}{2}\pm \frac {1}{2}\sqrt {2-p^{2}}\) | \(0\) | d’Alembert |