This article is about certain differential equations. For certain functional equations named after Abel, see
Abel equation .
In mathematics , an Abel equation of the first kind , named after Niels Henrik Abel , is any ordinary differential equation that is cubic in the unknown function. In other words, it is an equation of the form
y
′
=
f
3
(
x
)
y
3
+
f
2
(
x
)
y
2
+
f
1
(
x
)
y
+
f
0
(
x
)
{\displaystyle y'=f_{3}(x)y^{3}+f_{2}(x)y^{2}+f_{1}(x)y+f_{0}(x)\,}
where
f
3
(
x
)
≠
0
{\displaystyle f_{3}(x)\neq 0}
Properties
If
f
3
(
x
)
=
0
{\displaystyle f_{3}(x)=0}
f
0
(
x
)
=
0
{\displaystyle f_{0}(x)=0}
f
2
(
x
)
=
0
{\displaystyle f_{2}(x)=0}
f
0
(
x
)
=
0
{\displaystyle f_{0}(x)=0}
Bernoulli equation , while if
f
3
(
x
)
=
0
{\displaystyle f_{3}(x)=0}
Riccati equation .
Solution
The substitution
y
=
1
u
{\displaystyle y={\dfrac {1}{u}}}
Abel equation of the second kind , of the form
u
u
′
=
−
f
0
(
x
)
u
3
−
f
1
(
x
)
u
2
−
f
2
(
x
)
u
−
f
3
(
x
)
.
{\displaystyle uu'=-f_{0}(x)u^{3}-f_{1}(x)u^{2}-f_{2}(x)u-f_{3}(x).\,}
The substitution
ξ
=
∫
f
3
(
x
)
E
2
d
x
,
u
=
(
y
+
f
2
(
x
)
3
f
3
(
x
)
)
E
−
1
,
E
=
exp
(
∫
(
f
1
(
x
)
−
f
2
2
(
x
)
3
f
3
(
x
)
)
d
x
)
{\displaystyle {\begin{aligned}\xi &=\int f_{3}(x)E^{2}~dx,\\[6pt]u&=\left(y+{\dfrac {f_{2}(x)}{3f_{3}(x)}}\right)E^{-1},\\[6pt]E&=\exp \left(\int \left(f_{1}(x)-{\frac {f_{2}^{2}(x)}{3f_{3}(x)}}\right)~dx\right)\end{aligned}}}
brings the Abel equation of the first kind to the canonical form
u
′
=
u
3
+
ϕ
(
ξ
)
.
{\displaystyle u'=u^{3}+\phi (\xi ).\,}
Dimitrios E. Panayotounakos and Theodoros I. Zarmpoutis discovered an analytic method to solve the above equation in an implicit form.[ 1]
Notes
References
Panayotounakos, D.E.; Panayotounakou, N.D.; Vakakis, A.F.A (2002). "On the Solution of the Unforced Damped Duffing Oscillator with No Linear Stiffness Term". Nonlinear Dynamics . 28 : 1– 16. doi :10.1023/A:1014925032022 . S2CID 117115358 . Mancas, Stefan C.; Rosu, Haret C. (2013). "Integrable dissipative nonlinear second order differential equations via factorizations and Abel equations". Physics Letters A . 377 : 1434– 1438. arXiv :1212.3636 doi :10.1016/j.physleta.2013.04.024 . 
![{\displaystyle {\begin{aligned}\xi &=\int f_{3}(x)E^{2}~dx,\\[6pt]u&=\left(y+{\dfrac {f_{2}(x)}{3f_{3}(x)}}\right)E^{-1},\\[6pt]E&=\exp \left(\int \left(f_{1}(x)-{\frac {f_{2}^{2}(x)}{3f_{3}(x)}}\right)~dx\right)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bccf6491acaae57b32db86e663b6135328c6a89a)
