Problems 8501 to 8600

Table 2.173: Main lookup table. Sorted sequentially by problem number.


#	ODE	CAS classiﬁcation	Solved?	Maple	Mma	Sympy

8501	$\cos (x) y^{''} = y^{'}$	[[_2nd_order, _missing_y]]	✓	✓	✓	✓

8502	$y^{''} = x {y^{'}}^{2}$ i.c.	[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]	✓	✓	✓	✓

8503	$y^{''} = x {y^{'}}^{2}$ i.c.	[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]	✓	✓	✓	✓

8504	$y^{''} = - e^{- 2 y}$ i.c.	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓	✓	✓	✗

8505	$y^{''} = - e^{- 2 y}$ i.c.	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓	✓	✓	✗

8506	$2 y^{''} = \sin (2 y)$ i.c.	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓	✗	✗	✗

8507	$2 y^{''} = \sin (2 y)$ i.c.	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓	✗	✗	✗

8508	$x^{3} y^{''} - x^{2} y^{'} = - x^{2} + 3$	[[_2nd_order, _missing_y]]	✓	✓	✓	✓

8509	$y^{''} = {y^{'}}^{2}$	[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_xy]]	✓	✓	✓	✓

8510	$y^{''} = e^{x} {y^{'}}^{2}$	[[_2nd_order, _missing_y]]	✓	✓	✓	✓

8511	$2 y^{''} = {y^{'}}^{3} \sin (2 x)$	[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]	✓	✓	✓	✗

8512	$x^{2} y^{''} + {y^{'}}^{2} = 0$	[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]	✓	✓	✓	✓

8513	$y^{''} = {y^{'}}^{2} + 1$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]	✓	✓	✓	✓

8514	$y^{''} = {({y^{'}}^{2} + 1)}^{3 / 2}$	[[_2nd_order, _missing_x]]	✓	✓	✓	✗

8515	$y y^{''} = {y^{'}}^{2} (1 - y^{'} \sin (y) - y y^{'} \cos (y))$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_y_y1]]	✓	✓	✓	✗

8516	$(1 + y^{2}) y^{''} + {y^{'}}^{3} + y^{'} = 0$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓	✓	✓	✗

8517	${(y y^{''} + 1 + {y^{'}}^{2})}^{2} = {({y^{'}}^{2} + 1)}^{3}$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓	✓	✓	✗

8518	$x^{2} y^{''} = y^{'} (2 x - y^{'})$ i.c.	[[_2nd_order, _missing_y]]	✓	✓	✓	✓

8519	$x^{2} y^{''} = y^{'} (3 x - 2 y^{'})$	[[_2nd_order, _missing_y]]	✓	✓	✓	✓

8520	$x y^{''} = y^{'} (2 - 3 y^{'} x)$	[[_2nd_order, _missing_y], _Liouville, [_2nd_order, _reducible, _mu_xy]]	✓	✓	✓	✓

8521	$x^{4} y^{''} = y^{'} (y^{'} + x^{3})$ i.c.	[[_2nd_order, _missing_y]]	✓	✓	✓	✓

8522	$y^{''} = 2 x + {(x^{2} - y^{'})}^{2}$	[[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_xy]]	✓	✓	✓	✓

8523	${y^{''}}^{2} - 2 y^{''} + {y^{'}}^{2} - 2 y^{'} x + x^{2} = 0$ i.c.	[[_2nd_order, _missing_y]]	✓	✓	✗	✗

8524	${y^{''}}^{2} - x y^{''} + y^{'} = 0$	[[_2nd_order, _missing_y]]	✓	✓	✓	✓

8525	${y^{''}}^{3} = 12 y^{'} (x y^{''} - 2 y^{'})$	[[_2nd_order, _missing_y]]	✓	✓	✗	✗

8526	$3 y y^{'} y^{''} = {y^{'}}^{3} - 1$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓	✓	✓	✗

8527	$4 y {y^{'}}^{2} y^{''} = {y^{'}}^{4} + 3$	[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]	✓	✓	✓	✗

8528	$y^{''} + y = - \cos (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✓

8529	$y^{''} - 6 y^{'} + 9 y = e^{x}$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✓

8530	$y^{''} + 3 y^{'} + 2 y = 12 x^{2}$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✓

8531	$y^{''} + 3 y^{'} + 2 y = x^{2} + 2 x + 1$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✓

8532	$x^{3} {y^{'}}^{2} + x^{2} y y^{'} + 4 = 0$	[[_homogeneous, ‘class G‘], _rational]	✓	✓	✓	✗

8533	$6 x {y^{'}}^{2} - (3 x + 2 y) y^{'} + y = 0$	[_quadrature]	✓	✓	✓	✓

8534	$9 {y^{'}}^{2} + 3 x y^{4} y^{'} + y^{5} = 0$	[[_1st_order, _with_linear_symmetries]]	✓	✓	✓	✗

8535	$4 y^{3} {y^{'}}^{2} - 4 y^{'} x + y = 0$	[[_1st_order, _with_linear_symmetries], _rational]	✓	✓	✓	✗

8536	$x^{6} {y^{'}}^{2} - 2 y^{'} x - 4 y = 0$	[[_homogeneous, ‘class G‘], _rational]	✓	✓	✓	✓

8537	$5 {y^{'}}^{2} + 6 y^{'} x - 2 y = 0$	[[_1st_order, _with_linear_symmetries], _dAlembert]	✓	✓	✓	✗

8538	$y^{2} {y^{'}}^{2} - (x + 1) y y^{'} + x = 0$	[_quadrature]	✓	✓	✓	✓

8539	$4 x^{5} {y^{'}}^{2} + 12 x^{4} y y^{'} + 9 = 0$	[[_homogeneous, ‘class G‘]]	✓	✓	✓	✗

8540	$4 y^{2} {y^{'}}^{3} - 2 y^{'} x + y = 0$	[[_1st_order, _with_linear_symmetries]]	✓	✓	✓	✗

8541	${y^{'}}^{4} + y^{'} x - 3 y = 0$	[[_1st_order, _with_linear_symmetries], _dAlembert]	✓	✓	✗	✗

8542	$x^{2} {y^{'}}^{3} - 2 x y {y^{'}}^{2} + y^{2} y^{'} + 1 = 0$	[[_1st_order, _with_linear_symmetries], _Clairaut]	✓	✓	✓	✗

8543	$16 x {y^{'}}^{2} + 8 y y^{'} + y^{6} = 0$	[[_homogeneous, ‘class G‘]]	✓	✓	✓	✓

8544	$x {y^{'}}^{2} - (x^{2} + 1) y^{'} + x = 0$	[_quadrature]	✓	✓	✓	✓

8545	${y^{'}}^{3} - 2 y^{'} x - y = 0$	[[_1st_order, _with_linear_symmetries], _dAlembert]	✓	✓	✗	✗

8546	$9 x y^{4} {y^{'}}^{2} - 3 y^{5} y^{'} - 1 = 0$	[[_homogeneous, ‘class G‘], _rational]	✓	✓	✓	✗

8547	$x^{2} {y^{'}}^{2} - (1 + 2 x y) y^{'} + 1 + y^{2} = 0$	[[_1st_order, _with_linear_symmetries], _rational, _Clairaut]	✓	✓	✓	✗

8548	$x^{6} {y^{'}}^{2} = 8 y^{'} x + 16 y$	[[_homogeneous, ‘class G‘]]	✓	✓	✓	✓

8549	$x^{2} {y^{'}}^{2} = {(x - y)}^{2}$	[_linear]	✓	✓	✓	✓

8550	${(y^{'} + 1)}^{2} (y - y^{'} x) = 1$	[[_1st_order, _with_linear_symmetries], _dAlembert]	✓	✓	✓	✗

8551	${y^{'}}^{3} - {y^{'}}^{2} + y^{'} x - y = 0$	[[_1st_order, _with_linear_symmetries], _Clairaut]	✓	✓	✓	✗

8552	$x {y^{'}}^{2} + y (1 - x) y^{'} - y^{2} = 0$	[_quadrature]	✓	✓	✓	✓

8553	$y {y^{'}}^{2} - (x + y) y^{'} + y = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	✓	✓	✓

8554	$x {y^{'}}^{2} + (k - x - y) y^{'} + y = 0$	[[_1st_order, _with_linear_symmetries], _rational, _dAlembert]	✓	✓	✓	✗

8555	$x {y^{'}}^{3} - 2 y {y^{'}}^{2} + 4 x^{2} = 0$	[[_1st_order, _with_linear_symmetries]]	✓	✓	✓	✗

8556	$y^{''} + y = 0$	[[_2nd_order, _missing_x]]	✓	✓	✓	✓

8557	$y^{''} - 9 y = 0$	[[_2nd_order, _missing_x]]	✓	✓	✓	✓

8558	$y^{''} + 3 y^{'} x + 3 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	✓	✓	✓

8559	$(4 x^{2} + 1) y^{''} - 8 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	✓	✓	✓

8560	$(- 4 x^{2} + 1) y^{''} + 8 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	✓	✓	✓

8561	$(x^{2} + 1) y^{''} - 4 y^{'} x + 6 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✓

8562	$(x^{2} + 1) y^{''} + 10 y^{'} x + 20 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✓

8563	$(x^{2} + 4) y^{''} + 2 y^{'} x - 12 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✓

8564	$(x^{2} - 9) y^{''} + 3 y^{'} x - 3 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✓

8565	$y^{''} + 2 y^{'} x + 5 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✓

8566	$(x^{2} + 4) y^{''} + 6 y^{'} x + 4 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	✓	✓	✓

8567	$(2 x^{2} + 1) y^{''} - 5 y^{'} x + 3 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✓

8568	$y^{''} + x^{2} y = 0$	[[_Emden, _Fowler]]	✓	✓	✓	✓

8569	$(- 4 x^{2} + 1) y^{''} + 6 y^{'} x - 4 y = 0$	[_Gegenbauer]	✓	✓	✓	✓

8570	$(2 x^{2} + 1) y^{''} + 3 y^{'} x - 3 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✓

8571	$y^{'''} + x^{2} y^{''} + 5 y^{'} x + 3 y = 0$	[[_3rd_order, _exact, _linear, _homogeneous]]	✓	✓	✓	✗

8572	$y^{''} + y^{'} x + 3 y = x^{2}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	✓	✓	✗

8573	$y^{''} + 2 y^{'} x + 2 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	✓	✓	✓

8574	$y^{''} + 3 y^{'} x + 7 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✓

8575	$2 y^{''} + 9 y^{'} x - 36 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✓

8576	$(x^{2} + 4) y^{''} + y^{'} x - 9 y = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	✓	✓	✓

8577	$(x^{2} + 4) y^{''} + 3 y^{'} x - 8 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✓

8578	$(9 x^{2} + 1) y^{''} - 18 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	✓	✓	✓

8579	$(3 x^{2} + 1) y^{''} + 13 y^{'} x + 7 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	✓	✓	✓

8580	$(2 x^{2} + 1) y^{''} + 11 y^{'} x + 9 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✓

8581	$y^{''} - 2 (3 + x) y^{'} - 3 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✓

8582	$y^{''} + (x - 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✓

8583	$(x^{2} - 2 x + 2) y^{''} - 4 (x - 1) y^{'} + 6 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✓

8584	$2 x (x + 1) y^{''} + 3 (x + 1) y^{'} - y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	✓	✓	✓

8585	$4 x^{2} y^{''} + 4 y^{'} x + (4 x^{2} - 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✓

8586	$4 x^{2} y^{''} + 4 y^{'} x - (4 x^{2} + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✓

8587	$4 x y^{''} + 3 y^{'} + 3 y = 0$	[[_Emden, _Fowler]]	✓	✓	✓	✓

8588	$2 x^{2} (1 - x) y^{''} - x (1 + 7 x) y^{'} + y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✓

8589	$2 x y^{''} + 5 (- 2 x + 1) y^{'} - 5 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✓

8590	$8 x^{2} y^{''} + 10 y^{'} x - (x + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✓

8591	$2 x y^{''} + (2 - x) y^{'} - 2 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✓

8592	$2 x (3 + x) y^{''} - 3 (x + 1) y^{'} + 2 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✓

8593	$2 x y^{''} + (- 2 x^{2} + 1) y^{'} - 4 x y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	✓	✓	✓

8594	$x (4 - x) y^{''} + (2 - x) y^{'} + 4 y = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	✓	✓	✓

8595	$3 x^{2} y^{''} + y^{'} x - (x + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✓

8596	$2 x y^{''} + (2 x + 1) y^{'} + 4 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✓

8597	$2 x y^{''} + (2 x + 1) y^{'} - 5 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✓

8598	$2 x^{2} y^{''} - 3 x (1 - x) y^{'} + 2 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✓

8599	$2 x^{2} y^{''} + x (4 x - 1) y^{'} + 2 (3 x - 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✓

8600	$2 x y^{''} - (2 x^{2} + 1) y^{'} - x y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✓

2.2.86 Problems 8501 to 8600