second order linear exact ode

Table 2.461: second order linear exact ode


#	ODE	ODE classiﬁcation	Solved?

11	$x^{''} = 50$ i.c.	[[_2nd_order, _quadrature]]	✓

12	$x^{''} = - 20$ i.c.	[[_2nd_order, _quadrature]]	✓

13	$x^{''} = 3 t$ i.c.	[[_2nd_order, _quadrature]]	✓

14	$x^{''} = 2 t + 1$ i.c.	[[_2nd_order, _quadrature]]	✓

15	$x^{''} = 4 {(t + 3)}^{2}$ i.c.	[[_2nd_order, _quadrature]]	✓

16	$x^{''} = \frac{1}{\sqrt{t + 4}}$ i.c.	[[_2nd_order, _quadrature]]	✓

17	$x^{''} = \frac{1}{{(1 + t)}^{3}}$ i.c.	[[_2nd_order, _quadrature]]	✓

18	$x^{''} = 50 \sin (5 t)$ i.c.	[[_2nd_order, _quadrature]]	✓

147	$x y^{''} = y^{'}$	[[_2nd_order, _missing_y]]	✓

150	$x y^{''} + y^{'} = 4 x$	[[_2nd_order, _missing_y]]	✓

221	$y^{''} + y^{'} = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

236	$y^{''} + 5 y^{'} = 0$	[[_2nd_order, _missing_x]]	✓

237	$2 y^{''} + 3 y^{'} = 0$	[[_2nd_order, _missing_x]]	✓

244	$x^{2} y^{''} + y^{'} x - y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

272	$2 y^{''} - 3 y^{'} = 0$	[[_2nd_order, _missing_x]]	✓

376	$x^{2} y^{''} + y^{'} x - y = 72 x^{5}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

813	$y^{''} + y^{'} = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

825	$y^{''} + 5 y^{'} = 0$	[[_2nd_order, _missing_x]]	✓

826	$2 y^{''} + 3 y^{'} = 0$	[[_2nd_order, _missing_x]]	✓

833	$x^{2} y^{''} + y^{'} x - y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

846	$2 y^{''} - 3 y^{'} = 0$	[[_2nd_order, _missing_x]]	✓

902	$x^{2} y^{''} + y^{'} x - y = 72 x^{5}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

1253	$y^{''} + 5 y^{'} = 0$	[[_2nd_order, _missing_x]]	✓

1294	$t^{2} y^{''} + 4 y^{'} t + 2 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

1296	$t^{2} y^{''} - 4 y^{'} t - 6 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

1330	$t^{2} y^{''} + 3 y^{'} t + y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

1345	$t^{2} y^{''} - 2 y = 3 t^{2} - 1$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

1352	$t^{2} y^{''} + 7 y^{'} t + 5 y = t$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

1742	$(x^{2} - 1) y^{''} + 4 y^{'} x + 2 y = 0$ i.c.	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

1746	$x^{2} y^{''} + y^{'} x - y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

1754	$(x^{2} - 4) y^{''} + 4 y^{'} x + 2 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

1811	$x^{2} y^{''} + y^{'} x - y = 2 x^{2} + 2$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

1828	$x^{2} y^{''} - y^{'} x - 3 y = x^{3 / 2}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

1837	${(x - 1)}^{2} y^{''} + 4 y^{'} x + 2 y = 2 x$ i.c.	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

1839	$(x + 1) (2 x + 3) y^{''} + 2 (x + 2) y^{'} - 2 y = {(2 x + 3)}^{2}$ i.c.	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

2362	$2 t^{2} y^{''} + 3 y^{'} t - y = 0$ i.c.	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

2363	$y^{''} + y^{'} t + y = 0$ i.c.	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

2400	$t^{2} y^{''} + 3 y^{'} t + y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

2433	$2 t^{2} y^{''} + 3 y^{'} t - y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

2435	$t^{2} y^{''} + 3 y^{'} t + y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

2543	$2 t^{2} y^{''} + 3 y^{'} t - y = 0$ i.c.	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

2544	$y^{''} + y^{'} t + y = 0$ i.c.	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

2581	$t^{2} y^{''} + 3 y^{'} t + y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

2591	$t^{2} y^{''} - 2 y = t^{2}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

2607	$y^{''} + 2 y^{'} = 1 + t^{2} + e^{- 2 t}$	[[_2nd_order, _missing_y]]	✓

2629	$2 t^{2} y^{''} + 3 y^{'} t - y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

2631	$t^{2} y^{''} + 3 y^{'} t + y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

3089	$y^{''} = 0$	[[_2nd_order, _quadrature]]	✓

3141	$2 y^{''} + y^{'} = 8 \sin (2 x) + e^{- x}$ i.c.	[[_2nd_order, _missing_y]]	✓

3217	$y^{''} + 2 y^{'} = x^{3} \sin (2 x)$	[[_2nd_order, _missing_y]]	✓

3218	$y^{''} - y^{'} = x e^{2 x} \sin (x)$	[[_2nd_order, _missing_y]]	✓

3220	$y^{''} + 2 y^{'} = x^{2} e^{- x} \sin (x)$	[[_2nd_order, _missing_y]]	✓

3228	$x^{2} y^{''} + 3 y^{'} x + y = 1 - x$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

3244	$y^{''} = \cos (t)$	[[_2nd_order, _quadrature]]	✓

3249	$x y^{''} = x^{2} + 1$	[[_2nd_order, _quadrature]]	✓

3250	$(1 - x) y^{''} = y^{'}$	[[_2nd_order, _missing_y]]	✓

3251	$(x^{2} + 1) y^{''} + 2 x (y^{'} + 1) = 0$	[[_2nd_order, _missing_y]]	✓

3253	$x y^{''} + x = y^{'}$	[[_2nd_order, _missing_y]]	✓

3272	$y^{''} = \sec (x) \tan (x)$ i.c.	[[_2nd_order, _quadrature]]	✓

3284	$(1 - e^{x}) y^{''} = e^{x} y^{'}$ i.c.	[[_2nd_order, _missing_y]]	✓

3494	${(x + 1)}^{2} y^{''} + 3 (x + 1) y^{'} + y = x^{2}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

3565	$x^{2} y^{''} + 5 y^{'} x + 3 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

3575	$x^{2} y^{''} + y^{'} x - y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

3584	$y^{''} = x e^{x}$	[[_2nd_order, _quadrature]]	✓

3585	$y^{''} = x^{n}$	[[_2nd_order, _quadrature]]	✓

3587	$y^{''} = \cos (x)$ i.c.	[[_2nd_order, _quadrature]]	✓

3589	$y^{''} = x e^{x}$ i.c.	[[_2nd_order, _quadrature]]	✓

3699	$y^{''} + 4 y^{'} = 0$	[[_2nd_order, _missing_x]]	✓

3708	$2 x^{2} y^{''} + 5 y^{'} x + y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

3773	$x^{2} y^{''} + 4 y^{'} x + 2 y = 4 \ln (x)$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

3774	$x^{2} y^{''} + 4 y^{'} x + 2 y = \cos (x)$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

4124	$2 y^{''} + 3 y^{'} = 0$	[[_2nd_order, _missing_x]]	✓

4127	$y^{''} = 0$	[[_2nd_order, _quadrature]]	✓

4426	$x y^{''} = x + y^{'}$	[[_2nd_order, _missing_y]]	✓

4484	$y^{''} - y^{'} = e^{x} (x^{2} + 10)$	[[_2nd_order, _missing_y]]	✓

4508	$y^{''} + y^{'} = \frac{1}{e^{x} + 1}$	[[_2nd_order, _missing_y]]	✓

5916	$y^{''} + 2 y^{'} = 0$	[[_2nd_order, _missing_x]]	✓

5945	$y^{''} = 0$ i.c.	[[_2nd_order, _quadrature]]	✓

5958	$y^{''} - 3 y^{'} = 2 e^{2 x} \sin (x)$	[[_2nd_order, _missing_y]]	✓

5959	$y^{''} + y^{'} = x^{2} + 2 x$	[[_2nd_order, _missing_y]]	✓

5960	$y^{''} + y^{'} = x + \sin (2 x)$	[[_2nd_order, _missing_y]]	✓

5991	$y^{''} - \frac{2 y^{'}}{x} + \frac{2 y}{x^{2}} = x \ln (x)$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

5993	$x^{2} y^{''} + y^{'} x - y = x^{2} e^{- x}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

5994	$2 x^{2} y^{''} + 3 y^{'} x - y = \frac{1}{x}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

5999	$x y^{''} - y^{'} = x^{2}$	[[_2nd_order, _missing_y]]	✓

6009	$(x^{2} + 1) y^{''} + 2 x (y^{'} + 1) = 0$	[[_2nd_order, _missing_y]]	✓

6015	$x y^{''} - y^{'} = x^{2}$ i.c.	[[_2nd_order, _missing_y]]	✓

6137	$y^{''} + 9 y^{'} = 0$	[[_2nd_order, _missing_x]]	✓

6142	$y^{''} + 5 y^{'} = 0$	[[_2nd_order, _missing_x]]	✓

6151	$y^{''} - 4 y^{'} = 10$	[[_2nd_order, _missing_x]]	✓

6172	$2 y^{''} + y^{'} = 2 x$	[[_2nd_order, _missing_y]]	✓

6182	$y^{''} - 2 y^{'} = 9 x e^{- x} - 6 x^{2} + 4 e^{2 x}$	[[_2nd_order, _missing_y]]	✓

6197	$x^{2} y^{''} + y^{'} x - y = x - \frac{1}{x}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

6219	$x y^{''} + y^{'} = 4 x$	[[_2nd_order, _missing_y]]	✓

6408	$x (1 - x) y^{''} + 2 (- 2 x + 1) y^{'} - 2 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

6513	$y^{''} = 9 x^{2} + 2 x - 1$	[[_2nd_order, _quadrature]]	✓

6539	$y^{''} - 7 y^{'} = - 3$	[[_2nd_order, _missing_x]]	✓

6540	$y^{''} + \frac{y^{'}}{x} - \frac{y}{x^{2}} = \ln (x)$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

6712	$y^{''} - 4 y^{'} = 5$	[[_2nd_order, _missing_x]]	✓

6753	${(x + 1)}^{2} y^{''} + (x + 1) y^{'} - y = \ln {(x + 1)}^{2} + x - 1$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

6754	${(2 x + 1)}^{2} y^{''} - 2 (2 x + 1) y^{'} - 12 y = 6 x$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

6773	$(x^{2} + 1) y^{''} + 2 y^{'} x = \frac{2}{x^{3}}$	[[_2nd_order, _missing_y]]	✓

6774	$x y^{''} - y^{'} = - \frac{2}{x} - \ln (x)$	[[_2nd_order, _missing_y]]	✓

6910	$x y^{''} + 2 y^{'} = 0$	[[_2nd_order, _missing_y]]	✓

6927	$y^{''} = f (x)$	[[_2nd_order, _quadrature]]	✓

6987	$x y^{''} - y^{'} = 0$	[[_2nd_order, _missing_y]]	✓

6988	$y^{''} = y^{'}$	[[_2nd_order, _missing_x]]	✓

7479	$y^{''} + \frac{y^{'}}{x} - \frac{y}{x^{2}} = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

7484	$(- x^{2} + 1) y^{''} - y^{'} x + y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

7490	$y^{''} + y^{'} x + y = 2 x e^{x} - 1$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

7492	$x^{2} y^{''} + y^{'} x - y = x^{2} + 2 x$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

7494	$x (x + 1) y^{''} + (x + 2) y^{'} - y = x + \frac{1}{x}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

7581	$y^{''} = x + 2$	[[_2nd_order, _quadrature]]	✓

7589	$y^{''} = 3 x + 1$	[[_2nd_order, _quadrature]]	✓

7615	$y^{''} = 0$	[[_2nd_order, _quadrature]]	✓

7674	$y^{''} + \frac{y^{'}}{x} - \frac{y}{x^{2}} = 0$ i.c.	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

7675	$y^{''} + \frac{y^{'}}{x} - \frac{y}{x^{2}} = 0$ i.c.	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

7676	${(3 x - 1)}^{2} y^{''} + (9 x - 3) y^{'} - 9 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

7759	$y^{''} + y^{'} = 1$	[[_2nd_order, _missing_x]]	✓

7764	$x y^{''} - 2 y^{'} = x^{3}$	[[_2nd_order, _missing_y]]	✓

7911	$x y^{''} + y^{'} = 4 x$	[[_2nd_order, _missing_y]]	✓

7936	$x y^{''} - 3 y^{'} = 5 x$	[[_2nd_order, _missing_y]]	✓

7977	$y^{''} - 2 y^{'} = 12 x - 10$	[[_2nd_order, _missing_y]]	✓

7980	$y^{''} + y^{'} = 10 x^{4} + 2$	[[_2nd_order, _missing_y]]	✓

8053	$y^{''} = \tan (x)$ i.c.	[[_2nd_order, _quadrature]]	✓

8054	$y^{''} - 2 y^{'} = \ln (x)$ i.c.	[[_2nd_order, _missing_y]]	✓

8061	$x^{2} y^{''} + 3 y^{'} x + y = \frac{2}{x}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

8223	$y^{''} - y^{'} = 0$	[[_2nd_order, _missing_x]]	✓

8225	$y^{''} + 2 y^{'} = 0$	[[_2nd_order, _missing_x]]	✓

8362	$t y^{''} - y^{'} = 2 t^{2}$ i.c.	[[_2nd_order, _missing_y]]	✓

8496	$x y^{''} = y^{'} + x^{5}$ i.c.	[[_2nd_order, _missing_y]]	✓

8497	$x y^{''} + y^{'} + x = 0$ i.c.	[[_2nd_order, _missing_y]]	✓

8759	$t y^{''} + 4 y^{'} = t^{2}$	[[_2nd_order, _missing_y]]	✓

8760	$(t^{2} + 9) y^{''} + 2 y^{'} t = 0$ i.c.	[[_2nd_order, _missing_y]]	✓

8762	$t y^{''} + y^{'} = 0$	[[_2nd_order, _missing_y]]	✓

8766	$y^{''} = 0$	[[_2nd_order, _quadrature]]	✓

8767	$y^{''} = 1$	[[_2nd_order, _quadrature]]	✓

8768	$y^{''} = f (t)$	[[_2nd_order, _quadrature]]	✓

8769	$y^{''} = k$	[[_2nd_order, _quadrature]]	✓

8772	$y^{''} = 4 \sin (x) - 4$	[[_2nd_order, _quadrature]]	✓

9072	$y^{''} = 0$	[[_2nd_order, _quadrature]]	✓

9075	$a y^{''} = 0$	[[_2nd_order, _quadrature]]	✓

9078	$y^{''} = 1$	[[_2nd_order, _quadrature]]	✓

9080	$y^{''} = x$	[[_2nd_order, _quadrature]]	✓

9083	$y^{''} + y^{'} = 0$	[[_2nd_order, _missing_x]]	✓

9086	$y^{''} + y^{'} = 1$	[[_2nd_order, _missing_x]]	✓

9089	$y^{''} + y^{'} = x$	[[_2nd_order, _missing_y]]	✓

9102	$y^{''} + y^{'} = 1$	[[_2nd_order, _missing_x]]	✓

9103	$y^{''} + y^{'} = x$	[[_2nd_order, _missing_y]]	✓

9104	$y^{''} + y^{'} = x + 1$	[[_2nd_order, _missing_y]]	✓

9105	$y^{''} + y^{'} = x^{2} + x + 1$	[[_2nd_order, _missing_y]]	✓

9106	$y^{''} + y^{'} = x^{3} + x^{2} + x + 1$	[[_2nd_order, _missing_y]]	✓

9107	$y^{''} + y^{'} = \sin (x)$	[[_2nd_order, _missing_y]]	✓

9108	$y^{''} + y^{'} = \cos (x)$	[[_2nd_order, _missing_y]]	✓

10997	$y^{''} = 0$	[[_2nd_order, _quadrature]]	✓

11030	$y^{''} + y^{'} x + y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

11075	$x y^{''} + y^{'} = 0$	[[_2nd_order, _missing_y]]	✓

11091	$x y^{''} - y^{'} x - y - x (x + 1) e^{x} = 0$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

11141	$x^{2} y^{''} + y^{'} x - y - a x^{2} = 0$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

11147	$x^{2} y^{''} + (x + a) y^{'} - y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

11150	$x^{2} y^{''} + 2 y^{'} x = 0$	[[_2nd_order, _missing_y]]	✓

11157	$x^{2} y^{''} - 2 y^{'} x - 4 y - x \sin (x) - (a x^{2} + 12 a + 4) \cos (x) = 0$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

11163	$x^{2} y^{''} + (3 x - 1) y^{'} + y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

11165	$x^{2} y^{''} - 3 y^{'} x - 5 y - x^{2} \ln (x) = 0$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

11210	$(x^{2} + 1) y^{''} + 4 y^{'} x + 2 y - 2 \cos (x) + 2 x = 0$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

11211	$(x^{2} + 1) y^{''} + a x y^{'} + (a - 2) y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

11217	$(x^{2} - 1) y^{''} + 2 y^{'} x = 0$	[[_2nd_order, _missing_y]]	✓

11218	$(x^{2} - 1) y^{''} + 2 y^{'} x - a = 0$	[[_2nd_order, _missing_y]]	✓

11226	$(x^{2} - 1) y^{''} - 2 (v - 1) x y^{'} - 2 v y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

11233	$x (x + 1) y^{''} + (3 x + 2) y^{'} + y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

11235	$x (x - 1) y^{''} + a y^{'} - 2 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

11243	$x (3 + x) y^{''} + (3 x - 1) y^{'} + y - (20 x + 30) {(x^{2} + 3 x)}^{7 / 3} = 0$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

11246	${(x - 2)}^{2} y^{''} - (x - 2) y^{'} - 3 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

11247	$2 x^{2} y^{''} - (2 x^{2} + l - 5 x) y^{'} - (4 x - 1) y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

11264	${(2 x + 1)}^{2} y^{''} - 2 (2 x + 1) y^{'} - 12 y - 3 x - 1 = 0$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

11266	${(3 x - 1)}^{2} y^{''} + 3 (3 x - 1) y^{'} - 9 y - \ln {(3 x - 1)}^{2} = 0$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

11278	$(a^{2} x^{2} - 1) y^{''} + 2 a^{2} x y^{'} = 0$	[[_2nd_order, _missing_y]]	✓

11280	$(a x^{2} + b x) y^{''} + 2 b y^{'} - 2 a y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

11291	$x (x^{2} + 1) y^{''} + 2 (x^{2} - 1) y^{'} - 2 x y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

11298	$x (x^{2} + 2) y^{''} - y^{'} - 6 x y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

11408	$y^{''} = - \frac{\cos (x) y^{'}}{\sin (x)} + \frac{y}{\sin {(x)}^{2}}$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

12446	$y^{''} + (a x + b) y^{'} + a y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

12477	$y^{''} + (a x^{n} + b x^{m}) y^{'} + (a n x^{n - 1} + b m x^{m - 1}) y = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

12490	$x y^{''} + a x y^{'} + a y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

12503	$x y^{''} + (a x^{2} + b x + c) y^{'} + (2 a x + b) y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

12508	$x y^{''} + (a x^{2} + b) x y^{'} + (3 a x^{2} + b) y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

12515	$x y^{''} + (a x^{n} + b) y^{'} + a n x^{n - 1} y = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

12526	$(x + a) y^{''} + (b x + c) y^{'} + b y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

12530	$(x + γ) y^{''} + (a x^{n} + b x^{m} + c) y^{'} + (a n x^{n - 1} + b m x^{m - 1}) y = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

12584	$(x^{2} + a) y^{''} + 2 b x y^{'} + 2 (b - 1) y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

12596	$(a x^{2} + b x + c) y^{''} + (d x + k) y^{'} + (d - 2 a) y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

12610	$x (a x^{2} + b) y^{''} + 2 (a x^{2} + b) y^{'} - 2 y a x = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

12671	$(a x^{n} + b x + c) y^{''} = a n (n - 1) x^{n - 2} y$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓

12717	$y^{''} + (a e^{λ x} + b e^{μ x} + c) y^{'} + (a λ e^{λ x} + b μ e^{μ x}) y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

12722	$(a e^{λ x} + b) y^{''} - a λ^{2} e^{λ x} y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

12875	$y^{''} - 2 y^{'} = e^{2 x} + 1$	[[_2nd_order, _missing_y]]	✓

12879	$x^{2} y^{''} + 3 y^{'} x + y = \frac{1}{{(1 - x)}^{2}}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

12920	$y^{''} = x e^{x}$	[[_2nd_order, _quadrature]]	✓

12929	$x^{2} y^{''} + 3 y^{'} x + y = x$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓


12930	${(x - 1)}^{2} y^{''} + 4 (x - 1) y^{'} + 2 y = \cos (x)$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

12945	$(x^{2} - x) y^{''} + (2 + 4 x) y^{'} + 2 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

12952	$\sin (x) y^{''} - \cos (x) y^{'} + 2 y \sin (x) = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

12967	$x^{''} = - 3 \sqrt{t}$ i.c.	[[_2nd_order, _quadrature]]	✓

12972	$x^{'} + t x^{''} = 1$ i.c.	[[_2nd_order, _missing_y]]	✓

13025	$x^{''} + x^{'} = 3 t$	[[_2nd_order, _missing_y]]	✓

13042	$x^{''} - 2 x^{'} = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

13070	$x^{''} - x^{'} = 6 + e^{2 t}$	[[_2nd_order, _missing_y]]	✓

13078	$x^{''} - 2 x^{'} = 4$ i.c.	[[_2nd_order, _missing_x]]	✓

13085	$t^{2} x^{''} + 3 t x^{'} + x = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

13095	$t^{2} x^{''} - 2 x = t^{3}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

13178	$(x^{2} + 1) y^{''} + 4 y^{'} x + 2 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

13457	$(2 x + 1) (x + 1) y^{''} + 2 y^{'} x - 2 y = {(2 x + 1)}^{2}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

13475	$x^{2} y^{''} + 4 y^{'} x + 2 y = 4 \ln (x)$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

13481	$x^{2} y^{''} + 5 y^{'} x + 3 y = 0$ i.c.	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

13482	$x^{2} y^{''} - 2 y = 4 x - 8$ i.c.	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

13487	${(x + 2)}^{2} y^{''} - (x + 2) y^{'} - 3 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

13683	$y^{''} - 4 y^{'} = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

13695	$x^{''} - 4 x^{'} = t^{2}$	[[_2nd_order, _missing_y]]	✓

13722	$t^{2} x^{''} - 2 x = t^{3}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

13723	$x^{''} - 4 x^{'} = \tan (t)$	[[_2nd_order, _missing_y]]	✓

13728	$t^{2} x^{''} + t x^{'} - x = 0$ i.c.	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

13730	$x^{2} y^{''} - y^{'} x - 3 y = 0$ i.c.	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

13733	$3 x^{2} z^{''} + 5 x z^{'} - z = 0$ i.c.	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

13922	$y^{''} + 2 y^{'} x + 2 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

13923	$x y^{''} + y^{'} \sin (x) + \cos (x) y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

13924	$y^{''} + 2 x^{2} y^{'} + 4 x y = 2 x$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

13925	$(- x^{2} + 1) y^{''} + (1 - x) y^{'} + y = - 2 x + 1$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

13928	$y^{''} + x^{2} y^{'} + 2 x y = 2 x$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

13930	$x y^{''} + x^{2} y^{'} + 2 x y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

13931	$y^{''} + y^{'} \sin (x) + \cos (x) y = \cos (x)$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

13932	$y^{''} + \cot (x) y^{'} - y \csc {(x)}^{2} = \cos (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

13933	$x \ln (x) y^{''} + 2 y^{'} - \frac{y}{x} = 1$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

14016	$t^{2} y^{''} + 3 y^{'} t + y = t^{7}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

14159	$x y^{''} - y^{'} = x^{2} e^{x}$ i.c.	[[_2nd_order, _missing_y]]	✓

14190	$y^{''} - 3 y^{'} = 2 - 6 x$	[[_2nd_order, _missing_y]]	✓

14239	$x^{2} y^{''} + y^{'} x - y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

14241	$2 x^{2} y^{''} + 3 y^{'} x - y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

14243	$x^{2} y^{''} - 2 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

14257	$x^{2} y^{''} + 6 y^{'} x + 4 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

14856	$y^{''} + 2 y^{'} = 3 t + 2$ i.c.	[[_2nd_order, _missing_y]]	✓

14857	$y^{''} + 4 y^{'} = 3 t + 2$ i.c.	[[_2nd_order, _missing_y]]	✓

14913	$y^{''} = \frac{x + 1}{x - 1}$	[[_2nd_order, _quadrature]]	✓

14927	$y^{''} = \sin (2 x)$	[[_2nd_order, _quadrature]]	✓

14928	$y^{''} - 3 = x$	[[_2nd_order, _quadrature]]	✓

14936	$x y^{''} + 2 = \sqrt{x}$ i.c.	[[_2nd_order, _quadrature]]	✓

15138	$x y^{''} + 4 y^{'} = 18 x^{2}$	[[_2nd_order, _missing_y]]	✓

15139	$x y^{''} = 2 y^{'}$	[[_2nd_order, _missing_y]]	✓

15140	$y^{''} = y^{'}$	[[_2nd_order, _missing_x]]	✓

15141	$y^{''} + 2 y^{'} = 8 e^{2 x}$	[[_2nd_order, _missing_y]]	✓

15143	$(x^{2} + 1) y^{''} + 2 y^{'} x = 0$	[[_2nd_order, _missing_y]]	✓

15150	$y^{''} = 2 y^{'} - 6$	[[_2nd_order, _missing_x]]	✓

15152	$y^{''} + 4 y^{'} = 9 e^{- 3 x}$	[[_2nd_order, _missing_y]]	✓

15160	$y^{''} = y^{'}$	[[_2nd_order, _missing_x]]	✓

15166	$x y^{''} - y^{'} = 6 x^{5}$	[[_2nd_order, _missing_y]]	✓

15170	$y^{''} + 4 y^{'} = 9 e^{- 3 x}$	[[_2nd_order, _missing_y]]	✓

15172	$x y^{''} + 4 y^{'} = 18 x^{2}$ i.c.	[[_2nd_order, _missing_y]]	✓

15174	$y^{''} = y^{'}$ i.c.	[[_2nd_order, _missing_x]]	✓

15175	$y^{''} + 2 y^{'} = 8 e^{2 x}$ i.c.	[[_2nd_order, _missing_y]]	✓

15178	$x y^{''} + 2 y^{'} = 6$ i.c.	[[_2nd_order, _missing_y]]	✓

15241	$y^{''} + 5 y^{'} = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

15247	$y^{''} + 3 y^{'} = 0$	[[_2nd_order, _missing_x]]	✓

15309	$x^{2} y^{''} - 2 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

15320	$2 x^{2} y^{''} + 5 y^{'} x + y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

15364	$y^{''} + 3 y^{'} = e^{\frac{x}{2}}$	[[_2nd_order, _missing_y]]	✓

15368	$y^{''} + 3 y^{'} = 26 \cos (\frac{x}{3}) - 12 \sin (\frac{x}{3})$	[[_2nd_order, _missing_y]]	✓

15379	$y^{''} = 6 x e^{x} \sin (x)$	[[_2nd_order, _quadrature]]	✓

15384	$y^{''} + 4 y^{'} = 20$	[[_2nd_order, _missing_x]]	✓

15385	$y^{''} + 4 y^{'} = x^{2}$	[[_2nd_order, _missing_y]]	✓

15436	$2 x^{2} y^{''} + 5 y^{'} x + y = 85 \cos (2 \ln (x))$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

15437	$x^{2} y^{''} - 2 y = 15 \cos (3 \ln (x)) - 10 \sin (3 \ln (x))$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

15439	$2 x^{2} y^{''} + 5 y^{'} x + y = \frac{10}{x}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

15448	$x^{2} y^{''} + y^{'} x - y = \sqrt{x}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

15452	$x^{2} y^{''} - 2 y = \frac{1}{x - 2}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

15454	$x y^{''} + (2 x + 2) y^{'} + 2 y = 8 e^{2 x}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

15456	$x^{2} y^{''} - 2 y^{'} x - 4 y = \frac{10}{x}$ i.c.	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

15470	$2 x y^{''} + y^{'} = \sqrt{x}$	[[_2nd_order, _missing_y]]	✓

15490	$2 y^{''} - 7 y^{'} + 3 = 0$	[[_2nd_order, _missing_x]]	✓

15492	$x y^{''} = 3 y^{'}$	[[_2nd_order, _missing_y]]	✓

15493	$y^{''} - 5 y^{'} = 0$	[[_2nd_order, _missing_x]]	✓

15507	$x^{2} y^{''} + y^{'} x - y = \frac{1}{x^{2} + 1}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

15512	$x^{2} y^{''} + 3 y^{'} x + y = \frac{1}{{(x + 1)}^{2}}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

15513	$x^{2} y^{''} + 3 y^{'} x + y = \frac{1}{x}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

15723	$y^{''} + 9 y^{'} = 0$	[[_2nd_order, _missing_x]]	✓

15930	$y^{''} - \frac{y^{'}}{t} + \frac{y}{t^{2}} = \frac{1}{t}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

16118	$t^{2} y^{''} + y^{'} t - y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

16135	$y^{''} = 0$	[[_2nd_order, _quadrature]]	✓

16137	$y^{''} + y^{'} = 0$	[[_2nd_order, _missing_x]]	✓

16150	$y^{''} - y^{'} = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

16151	$3 y^{''} - y^{'} = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

16185	$y^{''} + 2 y^{'} = 3 - 4 t$	[[_2nd_order, _missing_y]]	✓

16190	$y^{''} = 3 t^{4} - 2 t$	[[_2nd_order, _quadrature]]	✓

16200	$y^{''} - 2 y^{'} = 52 \sin (3 t)$	[[_2nd_order, _missing_y]]	✓

16208	$y^{''} + 4 y^{'} = 8 e^{4 t} - 4 e^{- 4 t}$	[[_2nd_order, _missing_y]]	✓

16209	$y^{''} - 3 y^{'} = t^{2} - e^{3 t}$	[[_2nd_order, _missing_y]]	✓

16210	$y^{''} + 4 y^{'} = - 24 t - 6 - 4 t e^{- 4 t} + e^{- 4 t}$	[[_2nd_order, _missing_y]]	✓

16211	$y^{''} - 3 y^{'} = t^{2} - e^{3 t}$	[[_2nd_order, _missing_y]]	✓

16212	$y^{''} = t^{2} + e^{t} + \sin (t)$	[[_2nd_order, _quadrature]]	✓

16213	$y^{''} + 3 y^{'} = 18$ i.c.	[[_2nd_order, _missing_x]]	✓

16221	$y^{''} - 3 y^{'} = - e^{3 t} - 2 t$ i.c.	[[_2nd_order, _missing_y]]	✓

16222	$y^{''} - y^{'} = - 3 t - 4 e^{2 t} t^{2}$ i.c.	[[_2nd_order, _missing_y]]	✓

16223	$y^{''} - 2 y^{'} = 2 t^{2}$ i.c.	[[_2nd_order, _missing_y]]	✓

16224	$y^{''} + 4 y^{'} = - 24 t - 6 - 4 t e^{- 4 t} + e^{- 4 t}$ i.c.	[[_2nd_order, _missing_y]]	✓

16225	$y^{''} - 3 y^{'} = e^{- 3 t} - e^{3 t}$ i.c.	[[_2nd_order, _missing_y]]	✓

16238	$y^{''} + 16 y^{'} = t$	[[_2nd_order, _missing_y]]	✓

16283	$t^{2} y^{''} + 3 y^{'} t + y = \ln (t)$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

16285	$t^{2} y^{''} - 4 y^{'} t - 6 y = 2 \ln (t)$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

16407	$2 x^{2} y^{''} + 3 y^{'} x - y = \frac{1}{x^{2}}$ i.c.	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

16408	$x^{2} y^{''} + 4 y^{'} x + 2 y = \ln (x)$ i.c.	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

16412	$x^{2} y^{''} + 4 y^{'} x + 2 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

16425	$x^{2} y^{''} + 4 y^{'} x + 2 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

16505	$y^{''} + 5 y^{'} = 5 t^{2}$	[[_2nd_order, _missing_y]]	✓

16506	$y^{''} - 4 y^{'} = - 3 \sin (t)$	[[_2nd_order, _missing_y]]	✓

16509	$y^{''} - 2 y^{'} = \frac{1}{1 + e^{2 t}}$	[[_2nd_order, _missing_y]]	✓

16539	$2 x^{2} y^{''} + 5 y^{'} x + y = 0$ i.c.	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

16838	$(x - 1) y^{''} = 1$	[[_2nd_order, _quadrature]]	✓

16847	$y^{''} = x e^{x}$ i.c.	[[_2nd_order, _quadrature]]	✓

16848	$y^{''} = 2 x \ln (x)$	[[_2nd_order, _quadrature]]	✓

16849	$x y^{''} = y^{'}$	[[_2nd_order, _missing_y]]	✓

16850	$x y^{''} + y^{'} = 0$	[[_2nd_order, _missing_y]]	✓

16852	$x y^{''} = y^{'} + x^{2}$	[[_2nd_order, _missing_y]]	✓

16864	$y^{''} + y^{'} + 2 = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

16903	$y^{''} + 3 y^{'} = 3$	[[_2nd_order, _missing_x]]	✓

16904	$y^{''} - 7 y^{'} = {(x - 1)}^{2}$	[[_2nd_order, _missing_y]]	✓

16905	$y^{''} + 3 y^{'} = e^{x}$	[[_2nd_order, _missing_y]]	✓

16906	$y^{''} + 7 y^{'} = e^{- 7 x}$	[[_2nd_order, _missing_y]]	✓

16909	$4 y^{''} - 3 y^{'} = x e^{\frac{3 x}{4}}$	[[_2nd_order, _missing_y]]	✓

16910	$y^{''} - 4 y^{'} = x e^{4 x}$	[[_2nd_order, _missing_y]]	✓

16940	$y^{''} + 2 y^{'} = - 2$	[[_2nd_order, _missing_x]]	✓

16948	$y^{''} + 8 y^{'} = 8 x$	[[_2nd_order, _missing_y]]	✓

16952	$7 y^{''} - y^{'} = 14 x$	[[_2nd_order, _missing_y]]	✓

16953	$y^{''} + 3 y^{'} = 3 x e^{- 3 x}$	[[_2nd_order, _missing_y]]	✓

16962	$y^{''} - y^{'} = e^{x} \sin (x)$	[[_2nd_order, _missing_y]]	✓

16963	$y^{''} + 2 y^{'} = 4 e^{x} (\cos (x) + \sin (x))$	[[_2nd_order, _missing_y]]	✓

16965	$4 y^{''} + 8 y^{'} = x \sin (x)$	[[_2nd_order, _missing_y]]	✓

16980	$y^{''} + 4 y^{'} = x + e^{- 4 x}$	[[_2nd_order, _missing_y]]	✓

16986	$y^{''} - 4 y^{'} = 2 \cos {(4 x)}^{2}$	[[_2nd_order, _missing_y]]	✓

16988	$y^{''} - 3 y^{'} = 18 x - 10 \cos (x)$	[[_2nd_order, _missing_y]]	✓

16995	$y^{''} + y^{'} = \cos {(x)}^{2} + e^{x} + x^{2}$	[[_2nd_order, _missing_y]]	✓

16998	$y^{''} + y^{'} = x^{2} - e^{- x} + e^{x}$	[[_2nd_order, _missing_y]]	✓

17005	$y^{''} - 3 y^{'} = 1 + e^{x} + \cos (x) + \sin (x)$	[[_2nd_order, _missing_y]]	✓

17011	$y^{''} + 2 y^{'} + 1 = 3 \sin (2 x) + \cos (x)$	[[_2nd_order, _missing_y]]	✓

17023	$y^{''} + y^{'} = e^{- x}$ i.c.	[[_2nd_order, _missing_y]]	✓

17030	$y^{''} - y^{'} = - 5 e^{- x} (\cos (x) + \sin (x))$ i.c.	[[_2nd_order, _missing_y]]	✓

17046	$x^{2} y^{''} + y^{'} x - y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

17047	$x^{2} y^{''} + 3 y^{'} x + y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

17049	$x y^{''} + y^{'} = 0$	[[_2nd_order, _missing_y]]	✓

17057	$x^{2} y^{''} - 2 y = \sin (\ln (x))$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

17058	$x^{2} y^{''} - y^{'} x - 3 y = - \frac{16 \ln (x)}{x}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

17060	$x^{2} y^{''} + y^{'} x - y = x^{m}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

17061	$x^{2} y^{''} + 4 y^{'} x + 2 y = 2 \ln {(x)}^{2} + 12 x$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

17078	$y^{''} + y^{'} = \frac{1}{e^{x} + 1}$	[[_2nd_order, _missing_y]]	✓

17084	$y^{''} + y^{'} = e^{2 x} \cos (e^{x})$	[[_2nd_order, _missing_y]]	✓

17093	$(x^{2} + 1) y^{''} + 2 y^{'} x = \frac{1}{x^{2} + 1}$ i.c.	[[_2nd_order, _missing_y]]	✓

17117	$y^{''} + α y^{'} = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

17124	$x y^{''} + y^{'} = 0$	[[_2nd_order, _missing_y]]	✓

17494	$t^{2} y^{''} - 2 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

17524	$y^{''} + 5 y^{'} = 0$	[[_2nd_order, _missing_x]]	✓

17557	$x^{2} y^{''} + 4 y^{'} x + 2 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

17559	$x^{2} y^{''} - 4 y^{'} x - 6 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

17560	$x^{2} y^{''} - 2 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

17564	$2 x^{2} y^{''} + y^{'} x - 3 y = 0$ i.c.	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

17574	$y^{''} + 2 y^{'} = 3 + 4 \sin (2 t)$	[[_2nd_order, _missing_y]]	✓

17592	$y^{''} + 3 y^{'} = 2 t^{4} + t^{2} e^{- 3 t} + \sin (3 t)$	[[_2nd_order, _missing_y]]	✓

17602	$x^{2} y^{''} + 7 y^{'} x + 5 y = x$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

17634	$t^{2} y^{''} - 2 y = 3 t^{2} - 1$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

17637	$t^{2} y^{''} + 7 y^{'} t + 5 y = t$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

17822	$y^{''} = \sin (x)$	[[_2nd_order, _quadrature]]	✓

17958	$x^{2} y^{''} - 2 y = x^{2} + \frac{1}{x}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

18122	$x y^{''} + y^{'} = 4 x$	[[_2nd_order, _missing_y]]	✓

18178	$x y^{''} - y^{'} = 3 x^{2}$	[[_2nd_order, _missing_y]]	✓

18179	$x y^{''} + y^{'} = 0$	[[_2nd_order, _missing_y]]	✓

18182	$y^{''} - 2 y^{'} = 6$	[[_2nd_order, _missing_x]]	✓

18184	$y^{''} = e^{x}$	[[_2nd_order, _quadrature]]	✓

18185	$y^{''} - 2 y^{'} = 4$	[[_2nd_order, _missing_x]]	✓

18188	$y^{''} + 2 y^{'} = 6 e^{x}$	[[_2nd_order, _missing_y]]	✓

18189	$x^{2} y^{''} - 3 y^{'} x - 5 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

18195	$x^{2} y^{''} - 2 y = 0$ i.c.	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

18198	$y^{''} + y^{'} = 0$ i.c.	[[_2nd_order, _missing_x]]	✓

18225	$y^{''} + y^{'} = 0$	[[_2nd_order, _missing_x]]	✓

18259	$y^{''} - 2 y^{'} = 12 x - 10$	[[_2nd_order, _missing_y]]	✓

18262	$y^{''} + y^{'} = 10 x^{4} + 2$	[[_2nd_order, _missing_y]]	✓

18343	$y^{''} + y^{'} x + y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

18449	$x^{''} + 3 x^{'} = 0$	[[_2nd_order, _missing_x]]	✓

18524	$x^{2} y^{''} + 3 y^{'} x + y = \frac{1}{x}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

18607	$e y^{''} = \frac{P (\frac{L}{2} - x)}{2}$	[[_2nd_order, _quadrature]]	✓

18608	$e y^{''} = \frac{w (\frac{L^{2}}{4} - x^{2})}{2}$	[[_2nd_order, _quadrature]]	✓

18609	$e y^{''} = - \frac{(w L + P) x}{2} - \frac{w x^{2}}{2}$	[[_2nd_order, _quadrature]]	✓

18610	$e y^{''} = - P (L - x)$	[[_2nd_order, _quadrature]]	✓

18611	$e y^{''} = - P L + (w L + P) x - \frac{w (L^{2} + x^{2})}{2}$	[[_2nd_order, _quadrature]]	✓

18618	$x y^{''} + 2 y^{'} = 2 x$	[[_2nd_order, _missing_y]]	✓

18620	$(x^{2} - 1) y^{''} + 4 y^{'} x + 2 y = 2 x$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

18621	$(x^{2} + 1) y^{''} + 4 y^{'} x + 2 y = x$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

18622	$y^{''} - \cot (x) y^{'} + y \csc {(x)}^{2} = \cos (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓

18623	$(x^{2} - x) y^{''} + (3 x - 2) y^{'} + y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

18624	$(3 x^{2} + x) y^{''} + 2 (1 + 6 x) y^{'} + 6 y = \sin (x)$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

18627	$y^{''} = \cos (x)$	[[_2nd_order, _quadrature]]	✓

18634	$x y^{''} + 3 y^{'} = 3 x$	[[_2nd_order, _missing_y]]	✓

18635	$x = y^{''} + y^{'}$	[[_2nd_order, _missing_y]]	✓

18653	$y^{''} - \frac{2 y^{'}}{x} + \frac{2 y}{x^{2}} = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

18856	$x^{2} y^{''} - 2 y^{'} x - 4 y = x^{4}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

18859	$x^{2} y^{''} + 7 y^{'} x + 5 y = x^{5}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

18863	$x^{2} y^{''} + 4 y^{'} x + 2 y = e^{x}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

18869	$x^{2} y^{''} + y^{'} x - y = x^{m}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

18873	$x^{2} y^{''} + 3 y^{'} x + y = \frac{1}{{(1 - x)}^{2}}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

18878	$x y^{''} + 2 y^{'} x + 2 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

18879	$y^{''} + 2 e^{x} y^{'} + 2 y e^{x} = x^{2}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

18885	$y^{''} = x^{2} \sin (x)$	[[_2nd_order, _quadrature]]	✓

18903	$x y^{''} + y^{'} = 0$	[[_2nd_order, _missing_y]]	✓

18920	$\sin (x) y^{''} - \cos (x) y^{'} + 2 y \sin (x) = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

18922	$y^{''} = \frac{a}{x}$	[[_2nd_order, _quadrature]]	✓

18925	$y^{''} + y^{'} = e^{x}$	[[_2nd_order, _missing_y]]	✓

18928	$a y^{''} = y^{'}$	[[_2nd_order, _missing_x]]	✓

18932	$x y^{''} + (1 - x) y^{'} - y = e^{x}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

18935	$x^{2} y^{''} + y^{'} x - y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

18949	$(x^{2} + 1) y^{''} + 3 y^{'} x + y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

19255	$x^{2} y^{''} + 7 y^{'} x + 5 y = x^{5}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

19258	$x^{2} y^{''} - 2 y^{'} x - 4 y = x^{4}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

19259	$x^{2} y^{''} + y^{'} x - y = x^{m}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

19261	$x^{2} y^{''} + 2 y^{'} x = \ln (x)$	[[_2nd_order, _missing_y]]	✓

19262	$x^{2} y^{''} + 4 y^{'} x + 2 y = e^{x}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

19276	$x y^{''} + 2 y^{'} x + 2 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

19277	$y^{''} + e^{x} (y^{'} + y) = e^{x}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

19278	$(x^{2} + 1) y^{''} + 3 y^{'} x + y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

19281	$y^{''} + 2 e^{x} y^{'} + 2 y e^{x} = x^{2}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

19282	$(x^{2} - x) y^{''} + 2 (2 x + 1) y^{'} + 2 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

19283	$(x^{2} - x) y^{''} - 2 (x - 1) y^{'} - 4 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

19284	$(- x^{2} + 1) y^{''} - y^{'} x + y = 2 x$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

19285	$(2 x^{2} + 3 x) y^{''} + (3 + 6 x) y^{'} + 2 y = e^{x} (x + 1)$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

19287	$(- b x^{2} + a x) y^{''} + 2 a y^{'} + 2 b y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

19288	$\sin (x) y^{''} - \cos (x) y^{'} + 2 y \sin (x) = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

19295	$y^{''} = \sin (x) + x$	[[_2nd_order, _quadrature]]	✓

19296	$y^{''} = x e^{x}$	[[_2nd_order, _quadrature]]	✓

19299	$y^{''} = \frac{a}{x}$	[[_2nd_order, _quadrature]]	✓

19311	$y^{''} + y^{'} = e^{x}$	[[_2nd_order, _missing_y]]	✓

19320	$x y^{''} + y^{'} = x$	[[_2nd_order, _missing_y]]	✓

19329	$a y^{''} = y^{'}$	[[_2nd_order, _missing_x]]	✓

19356	$(x^{2} + 1) y^{''} + 3 y^{'} x + y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

19366	$x y^{''} + (1 - x) y^{'} = y + e^{x}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

19419	$x^{2} y^{''} + y^{'} x - y = 8 x^{3}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

19425	$x^{2} y^{''} + y^{'} x - y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓

19508	$x^{2} y^{''} - 2 y = x^{2} + \frac{1}{x}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

19514	$x^{2} y^{''} + 3 y^{'} x + y = \frac{1}{{(1 - x)}^{2}}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

19532	$y^{''} = x^{2} \sin (x)$	[[_2nd_order, _quadrature]]	✓

19533	$y^{''} = \sec {(x)}^{2}$	[[_2nd_order, _quadrature]]	✓

19539	$x y^{''} + y^{'} = 0$	[[_2nd_order, _missing_y]]	✓

19564	$x^{2} y^{''} + y^{'} x - y = x^{2} e^{x}$	[[_2nd_order, _exact, _linear, _nonhomogeneous]]	✓

2.4.5 second order linear exact ode