second order ode can be made integrable

Table 2.471: second order ode can be made integrable


#	ODE	ODE classiﬁcation	Solved?

149	$y^{''} + 4 y = 0$	[[_2nd_order, _missing_x]]	✓

216	$y^{''} - 9 y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

217	$y^{''} + 4 y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

218	$y^{''} + 25 y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

271	$y^{''} - 4 y = 0$	[[_2nd_order, _missing_x]]	✓

311	$y^{''} = (- 2 + 2 i \sqrt{3}) y$	[[_2nd_order, _missing_x]]	✓

808	$y^{''} - 9 y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

809	$y^{''} + 4 y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

810	$y^{''} + 25 y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

842	$y^{''} + 2 y = 4$	[[_2nd_order, _missing_x]]	✓

845	$y^{''} - 4 y = 0$	[[_2nd_order, _missing_x]]	✓

859	$y^{''} = (- 2 + 2 i \sqrt{3}) y$	[[_2nd_order, _missing_x]]	✓

1254	$4 y^{''} - 9 y = 0$	[[_2nd_order, _missing_x]]	✓

1264	$4 y^{''} - y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

1265	$y^{''} - y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

1268	$4 y^{''} - y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

1278	$4 y^{''} + 9 y = 0$	[[_2nd_order, _missing_x]]	✓

1286	$y^{''} + y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

1355	$u^{''} + 2 u = 0$	[[_2nd_order, _missing_x]]	✓

2364	$y^{''} - y = 0$	[[_2nd_order, _missing_x]]	✓

2545	$y^{''} - y = 0$	[[_2nd_order, _missing_x]]	✓

2564	$y^{''} + w^{2} y = 0$	[[_2nd_order, _missing_x]]	✓

2820	$z^{''} + z^{3} = 0$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓

2821	$z^{''} + z + z^{5} = 0$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓

2822	$z^{''} + e^{z^{2}} = 1$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓

2823	$z^{''} + \frac{z}{1 + z^{2}} = 0$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓

2824	$z^{''} + z - 2 z^{3} = 0$	[[_2nd_order, _missing_x], _Duﬃng, [_2nd_order, _reducible, _mu_x_y1]]	✓

2835	$y^{''} + λ y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

2836	$y^{''} + λ y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

2837	$y^{''} - λ y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

2838	$y^{''} + λ y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

2840	$y^{''} + λ y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

3059	$y^{''} - 4 y = 0$	[[_2nd_order, _missing_x]]	✓

3245	$y^{''} = k^{2} y$	[[_2nd_order, _missing_x]]	✓

3246	$x^{''} + k^{2} x = 0$	[[_2nd_order, _missing_x]]	✓

3266	$y^{''} = y$	[[_2nd_order, _missing_x]]	✓

3558	$y^{''} - 25 y = 0$	[[_2nd_order, _missing_x]]	✓

3559	$y^{''} + 4 y = 0$	[[_2nd_order, _missing_x]]	✓

3564	$y^{''} - 9 y = 0$	[[_2nd_order, _missing_x]]	✓

3698	$y^{''} - 36 y = 0$	[[_2nd_order, _missing_x]]	✓

4125	$y^{''} + 25 y = 0$	[[_2nd_order, _missing_x]]	✓

5918	$y^{''} - y = 0$	[[_2nd_order, _missing_x]]	✓

6002	$y^{''} = \frac{3 k y^{2}}{2}$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓

6003	$y^{''} = 2 k y^{3}$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓

6140	$y^{''} + 16 y = 0$	[[_2nd_order, _missing_x]]	✓

6243	$y^{''} = - 4 y$	[[_2nd_order, _missing_x]]	✓

6245	$y^{''} = y$	[[_2nd_order, _missing_x]]	✓

6388	$x^{''} - ω^{2} x = 0$	[[_2nd_order, _missing_x]]	✓

6504	$\frac{x^{''}}{2} = - 48 x$ i.c.	[[_2nd_order, _missing_x]]	✓

6575	$y^{''} - y = 0$	[[_2nd_order, _missing_x]]	✓

6706	$y^{''} + 25 y = 0$	[[_2nd_order, _missing_x]]	✓

6939	$x^{''} + x = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

6941	$x^{''} + x = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

6942	$x^{''} + x = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

6943	$y^{''} - y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

6946	$y^{''} - y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

6972	$y^{''} + 4 y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

6973	$y^{''} + 4 y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

6974	$y^{''} + 4 y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

6986	$y^{''} + 9 y = 18$	[[_2nd_order, _missing_x]]	✓

7008	$y^{''} + 9 y = 5$	[[_2nd_order, _missing_x]]	✓

7585	$y^{''} - y = 0$	[[_2nd_order, _missing_x]]	✓

7586	$y^{''} + 4 y = 0$	[[_2nd_order, _missing_x]]	✓

7587	$y^{''} + k^{2} y = 0$	[[_2nd_order, _missing_x]]	✓

7612	$y^{''} - 4 y = 0$	[[_2nd_order, _missing_x]]	✓

7613	$3 y^{''} + 2 y = 0$	[[_2nd_order, _missing_x]]	✓

7614	$y^{''} + 16 y = 0$	[[_2nd_order, _missing_x]]	✓

7621	$y^{''} + y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

7622	$y^{''} + y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

7623	$y^{''} + y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

7624	$y^{''} + y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

7628	$y^{''} + 10 y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

7650	$y^{''} + y = 0$	[[_2nd_order, _missing_x]]	✓

7651	$y^{''} - y = 0$	[[_2nd_order, _missing_x]]	✓

7762	$y^{''} + k^{2} y = 0$	[[_2nd_order, _missing_x]]	✓

7777	$y^{''} + 4 y = 0$	[[_2nd_order, _missing_x]]	✓

7778	$y^{''} - 4 y = 0$	[[_2nd_order, _missing_x]]	✓

7907	$y^{''} - k^{2} y = 0$	[[_2nd_order, _missing_x]]	✓

7939	$y^{''} + 8 y = 0$	[[_2nd_order, _missing_x]]	✓

7945	$y^{''} + y = 0$	[[_2nd_order, _missing_x]]	✓

7949	$y^{''} = 4 y$	[[_2nd_order, _missing_x]]	✓

8047	$y^{''} + 9 y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

8070	$y^{''} = - 3 y$ i.c.	[[_2nd_order, _missing_x]]	✓

8071	$y^{''} + \sin (y) = 0$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓

8219	$y^{''} + y = 0$	[[_2nd_order, _missing_x]]	✓

8221	$y^{''} - y = 0$	[[_2nd_order, _missing_x]]	✓

8307	$y^{''} + y = 0$	[[_2nd_order, _missing_x]]	✓

8500	$y^{''} + β^{2} y = 0$	[[_2nd_order, _missing_x]]	✓

8962	$y^{''} = A y^{2 / 3}$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓

8983	$y^{''} + e^{y} = 0$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓

9109	$y^{''} + y = 1$	[[_2nd_order, _missing_x]]	✓

9128	$y^{'} y^{''} + y^{2} = 0$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓

9129	$y^{'} y^{''} + y^{n} = 0$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓

10998	$y^{''} + y = 0$	[[_2nd_order, _missing_x]]	✓

11002	$y^{''} - y = 0$	[[_2nd_order, _missing_x]]	✓

11005	$y^{''} + l y = 0$	[[_2nd_order, _missing_x]]	✓

11551	$y^{''} - y^{2} = 0$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓

11552	$y^{''} - 6 y^{2} = 0$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓

11554	$y^{''} - 6 y^{2} + 4 y = 0$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓

11557	$y^{''} - a y^{3} = 0$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓

11560	$y^{''} + d + b y^{2} + c y + a y^{3} = 0$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓

11562	$y^{''} + 6 a^{10} y^{11} - y = 0$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓

11564	$y^{''} - e^{y} = 0$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓

11567	$y^{''} + a \sin (y) = 0$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓

12422	$y^{''} + a y = 0$	[[_2nd_order, _missing_x]]	✓

12924	$2 y^{''} = e^{y}$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓

13051	$x^{''} + 9 x = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

13052	$x^{''} - 12 x = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

13346	$y^{''} + 9 y = 0$	[[_2nd_order, _missing_x]]	✓

13347	$4 y^{''} + y = 0$	[[_2nd_order, _missing_x]]	✓

13607	$y^{''} + λ y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

13608	$y^{''} + λ y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

13609	$y^{''} + λ y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

13610	$y^{''} + λ y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

13691	$y^{''} - 4 y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

14078	$y^{''} + α^{2} y = 0$	[[_2nd_order, _missing_x]]	✓

14079	$y^{''} - α^{2} y = 0$	[[_2nd_order, _missing_x]]	✓

14157	$y^{''} = a^{2} y$	[[_2nd_order, _missing_x]]	✓

14166	$y^{''} = 9 y$	[[_2nd_order, _missing_x]]	✓

14167	$y^{''} + y = 0$	[[_2nd_order, _missing_x]]	✓

14168	$y^{''} - y = 0$	[[_2nd_order, _missing_x]]	✓

14233	$x^{''} + x - x^{3} = 0$	[[_2nd_order, _missing_x], _Duﬃng, [_2nd_order, _reducible, _mu_x_y1]]	✓

14234	$x^{''} + x + x^{3} = 0$	[[_2nd_order, _missing_x], _Duﬃng, [_2nd_order, _reducible, _mu_x_y1]]	✓

14237	$x^{''} = (2 \cos (x) - 1) \sin (x)$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓

14266	$y^{''} - y = 0$	[[_2nd_order, _missing_x]]	✓

14415	$y^{''} - y = 0$	[[_2nd_order, _missing_x]]	✓

14416	$y^{''} + y = 0$	[[_2nd_order, _missing_x]]	✓

14435	$y^{''} + α y = 0$	[[_2nd_order, _missing_x]]	✓

14825	$y^{''} + 2 y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

15225	$y^{''} + 4 y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

15238	$y^{''} - 4 y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

15246	$y^{''} - 25 y = 0$	[[_2nd_order, _missing_x]]	✓

15248	$4 y^{''} - y = 0$	[[_2nd_order, _missing_x]]	✓

15253	$y^{''} - 9 y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

15255	$y^{''} - 9 y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

15268	$y^{''} + 25 y = 0$	[[_2nd_order, _missing_x]]	✓

15273	$4 y^{''} + y = 0$	[[_2nd_order, _missing_x]]	✓

15274	$y^{''} + 16 y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

15276	$y^{''} + 16 y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

15464	$y^{''} + 36 y = 0$	[[_2nd_order, _missing_x]]	✓

15467	$y^{''} - 36 y = 0$	[[_2nd_order, _missing_x]]	✓

15473	$y^{''} + 3 y = 0$	[[_2nd_order, _missing_x]]	✓

16106	$y^{''} - y = 0$	[[_2nd_order, _missing_x]]	✓

16109	$y^{''} + 9 y = 0$	[[_2nd_order, _missing_x]]	✓

16111	$y^{''} + 9 y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

16116	$y^{''} + 16 y = 0$	[[_2nd_order, _missing_x]]	✓

16142	$4 y^{''} + 9 y = 0$	[[_2nd_order, _missing_x]]	✓

16143	$y^{''} + 16 y = 0$	[[_2nd_order, _missing_x]]	✓

16144	$y^{''} + 8 y = 0$	[[_2nd_order, _missing_x]]	✓

16145	$y^{''} + 7 y = 0$	[[_2nd_order, _missing_x]]	✓

16156	$y^{''} + 36 y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

16157	$y^{''} + 100 y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

16175	$y^{''} - 16 y = 0$	[[_2nd_order, _missing_x]]	✓

16237	$y^{''} + 4 y = 1$	[[_2nd_order, _missing_x]]	✓

16521	$y^{''} + 25 y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

16552	$4 x^{''} + 9 x = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

16553	$9 x^{''} + 4 x = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

16554	$x^{''} + 64 x = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

16555	$x^{''} + 100 x = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

16556	$x^{''} + x = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

16557	$x^{''} + 4 x = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

16558	$x^{''} + 16 x = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

16559	$x^{''} + 256 x = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

16560	$x^{''} + 9 x = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

16561	$10 x^{''} + \frac{x}{10} = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

16840	$y^{''} + y = 0$	[[_2nd_order, _missing_x]]	✓

16881	$y^{''} - y = 0$	[[_2nd_order, _missing_x]]	✓

16918	$y^{''} + k^{2} y = k$	[[_2nd_order, _missing_x]]	✓

16941	$y^{''} + 9 y = 9$	[[_2nd_order, _missing_x]]	✓

17038	$y^{''} - y = 1$	[[_2nd_order, _missing_x]]	✓

17109	$y^{''} + λ y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

17110	$y^{''} + λ y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

17111	$y^{''} - y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

17115	$y^{''} - y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

17120	$y^{''} + λ^{2} y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

17121	$y^{''} + λ^{2} y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

17481	$y^{''} + y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

17482	$y^{''} + 9 y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

17496	$y^{''} + 4 y = 0$	[[_2nd_order, _missing_x]]	✓

17525	$4 y^{''} - 9 y = 0$	[[_2nd_order, _missing_x]]	✓

17534	$4 y^{''} + 9 y = 0$	[[_2nd_order, _missing_x]]	✓

17547	$y^{''} + y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

17554	$4 y^{''} - y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

17570	$m y^{''} + k y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

17934	$y^{''} + y = 0$	[[_2nd_order, _missing_x]]	✓

17989	$y^{''} + 4 y = 0$	[[_2nd_order, _missing_x]]	✓

17990	$y^{''} - 4 y = 0$	[[_2nd_order, _missing_x]]	✓

18118	$y^{''} - k y = 0$	[[_2nd_order, _missing_x]]	✓

18191	$y^{''} - y = 0$	[[_2nd_order, _missing_x]]	✓

18219	$y^{''} + 8 y = 0$	[[_2nd_order, _missing_x]]	✓

18229	$y^{''} = 4 y$	[[_2nd_order, _missing_x]]	✓

18494	$θ^{''} = - p^{2} θ$	[[_2nd_order, _missing_x]]	✓

18509	$θ^{''} - p^{2} θ = 0$	[[_2nd_order, _missing_x]]	✓

18510	$y^{''} + y = 0$	[[_2nd_order, _missing_x]]	✓

18512	$r^{''} - a^{2} r = 0$	[[_2nd_order, _missing_x]]	✓

18528	$y^{''} = - m^{2} y$	[[_2nd_order, _missing_x]]	✓

18612	$e y^{''} = P (- y + a)$	[[_2nd_order, _missing_x]]	✓

18629	$y^{''} = - a^{2} y$	[[_2nd_order, _missing_x]]	✓

18655	$y^{''} - k^{2} y = 0$	[[_2nd_order, _missing_x]]	✓


18797	$y^{''} - m^{2} y = 0$	[[_2nd_order, _missing_x]]	✓

18886	$y^{''} + a^{2} y = 0$	[[_2nd_order, _missing_x]]	✓

19088	$y^{''} - n^{2} y = 0$	[[_2nd_order, _missing_x]]	✓

19303	$y^{''} = y$	[[_2nd_order, _missing_x]]	✓

19305	$y^{''} - a^{2} y = 0$	[[_2nd_order, _missing_x]]	✓

19307	$y^{''} = y^{3} - y$	[[_2nd_order, _missing_x], _Duﬃng, [_2nd_order, _reducible, _mu_x_y1]]	✓

19308	$y^{''} = e^{2 y}$ i.c.	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓

19350	$y^{''} = e^{y}$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓

19351	$y^{''} + a^{2} y = 0$	[[_2nd_order, _missing_x]]	✓

2.4.10 second order ode can be made integrable