4.4.12 Problems 1101 to 1200

Table 4.567: Second ODE homogeneous ODE

#

ODE

Mathematica

Maple

Sympy

6481

\[ {} 2 y y^{\prime \prime } = y^{2} \left (a x +b y\right )+{y^{\prime }}^{2} \]

6482

\[ {} 2 y y^{\prime \prime } = 3 y^{4}+{y^{\prime }}^{2} \]

6484

\[ {} 2 y y^{\prime \prime } = 8 y^{3}-2 y^{2} \left (f \left (x \right )^{2}+f^{\prime }\left (x \right )\right )-3 f \left (x \right ) y y^{\prime }+{y^{\prime }}^{2} \]

6486

\[ {} 2 y y^{\prime \prime } = 3 {y^{\prime }}^{2} \]

6487

\[ {} 2 y y^{\prime \prime } = 4 y^{2}+3 {y^{\prime }}^{2} \]

6488

\[ {} 2 y y^{\prime \prime } = f \left (x \right ) y^{2}+3 {y^{\prime }}^{2} \]

6489

\[ {} 2 y y^{\prime \prime } = y^{2} \left (1-3 y^{2}\right )+6 {y^{\prime }}^{2} \]

6490

\[ {} 2 y y^{\prime \prime } = -y^{2} \left (1+a y^{3}\right )+6 {y^{\prime }}^{2} \]

6491

\[ {} 2 y y^{\prime \prime } = {y^{\prime }}^{2} \left (1+{y^{\prime }}^{2}\right ) \]

6492

\[ {} 3 y y^{\prime \prime } = 36 y^{2}+2 {y^{\prime }}^{2} \]

6493

\[ {} 3 y y^{\prime \prime } = 5 {y^{\prime }}^{2} \]

6494

\[ {} 4 y y^{\prime \prime } = -4 y+3 {y^{\prime }}^{2} \]

6495

\[ {} 4 y y^{\prime \prime } = 12 y^{2}+3 {y^{\prime }}^{2} \]

6496

\[ {} 4 y y^{\prime \prime } = a y+b y^{2}+c y^{3}+3 {y^{\prime }}^{2} \]

6497

\[ {} 5 y y^{\prime \prime } = {y^{\prime }}^{2} \]

6498

\[ {} 12 y y^{\prime \prime } = -8 y^{3}+15 {y^{\prime }}^{2} \]

6499

\[ {} a y y^{\prime \prime } = \left (a -1\right ) {y^{\prime }}^{2} \]

6500

\[ {} a \left (2+a \right )^{2} y y^{\prime \prime } = -a^{2} f \left (x \right )^{2} y^{4}+a^{2} \left (2+a \right ) y^{3} f^{\prime }\left (x \right )+a \left (2+a \right )^{2} f \left (x \right ) y^{2} y^{\prime }+\left (a -1\right ) \left (2+a \right )^{2} {y^{\prime }}^{2} \]

6501

\[ {} y y^{\prime }+x {y^{\prime }}^{2}+x y y^{\prime \prime } = 0 \]

6502

\[ {} x {y^{\prime }}^{2}+x y y^{\prime \prime } = y y^{\prime } \]

6503

\[ {} x y y^{\prime \prime } = -y y^{\prime }+x {y^{\prime }}^{2} \]

6504

\[ {} x y y^{\prime \prime } = y \left (\operatorname {a2} +\operatorname {a3} y^{2}\right )+x \left (\operatorname {a0} +\operatorname {a1} y^{4}\right )-y y^{\prime }+x {y^{\prime }}^{2} \]

6505

\[ {} 2 y y^{\prime }+x {y^{\prime }}^{2}+x y y^{\prime \prime } = 0 \]

6506

\[ {} x {y^{\prime }}^{2}+x y y^{\prime \prime } = 3 y y^{\prime } \]

6508

\[ {} x y y^{\prime \prime } = x y^{3}+a y y^{\prime }+x {y^{\prime }}^{2} \]

6509

\[ {} x y y^{\prime \prime } = b^{2} x y^{3}+a y y^{\prime }+x {y^{\prime }}^{2} \]

6510

\[ {} y y^{\prime }+2 x {y^{\prime }}^{2}+x y y^{\prime \prime } = 0 \]

6511

\[ {} y y^{\prime }-2 x {y^{\prime }}^{2}+x y y^{\prime \prime } = 0 \]

6512

\[ {} -y y^{\prime }-2 x {y^{\prime }}^{2}+x y y^{\prime \prime } = 0 \]

6513

\[ {} x y y^{\prime \prime } = -\left (1+y\right ) y^{\prime }+2 x {y^{\prime }}^{2} \]

6514

\[ {} a y y^{\prime }+2 x {y^{\prime }}^{2}+x y y^{\prime \prime } = 0 \]

6515

\[ {} a y y^{\prime }-2 x {y^{\prime }}^{2}+x y y^{\prime \prime } = 0 \]

6516

\[ {} 4 y y^{\prime }-4 x {y^{\prime }}^{2}+x y y^{\prime \prime } = 0 \]

6517

\[ {} a y^{\prime } \left (x y^{\prime }-y\right )+x y y^{\prime \prime } = 0 \]

6518

\[ {} \left (x -y\right ) y^{\prime }+x {y^{\prime }}^{2}+x \left (x +y\right ) y^{\prime \prime } = y \]

6519

\[ {} 2 x y y^{\prime \prime } = -y y^{\prime }+x {y^{\prime }}^{2} \]

6521

\[ {} \left (x y^{\prime }-y\right )^{2}+x^{2} y y^{\prime \prime } = 0 \]

6522

\[ {} \left (x y^{\prime }-y\right )^{2}+x^{2} y y^{\prime \prime } = 3 y^{2} \]

6523

\[ {} x^{2} y y^{\prime \prime } = a y^{2}+a x y y^{\prime }+2 x^{2} {y^{\prime }}^{2} \]

6524

\[ {} c y^{2}+b x y y^{\prime }+a \,x^{2} {y^{\prime }}^{2}+x^{2} y y^{\prime \prime } = 0 \]

6525

\[ {} 2 \left (1-y\right )^{2} y-2 x \left (1-y\right ) y^{\prime }+2 x^{2} {y^{\prime }}^{2}+x^{2} \left (1-y\right ) y^{\prime \prime } = 0 \]

6526

\[ {} x^{2} \left (x -y\right ) y^{\prime \prime } = \left (x y^{\prime }-y\right )^{2} \]

6527

\[ {} \left (x y^{\prime }-y\right )^{2}+x^{2} \left (x -y\right ) y^{\prime \prime } = 0 \]

6528

\[ {} x^{2} \left (x -y\right ) y^{\prime \prime } = a \left (x y^{\prime }-y\right )^{2} \]

6529

\[ {} 2 x^{2} y y^{\prime \prime } = x^{2} {y^{\prime }}^{2}-y^{2} \]

6530

\[ {} 2 x^{2} y y^{\prime \prime } = -4 y^{2}+2 y y^{\prime } x +x^{2} {y^{\prime }}^{2} \]

6532

\[ {} x \left (1+x \right )^{2} y y^{\prime \prime } = a \left (x +2\right ) y^{2}-2 \left (x^{2}+1\right ) y y^{\prime }+x \left (1+x \right )^{2} {y^{\prime }}^{2} \]

6533

\[ {} 3 x y^{2}-12 x^{2} y y^{\prime }+4 \left (-x^{3}+1\right ) {y^{\prime }}^{2}+8 \left (-x^{3}+1\right ) y y^{\prime \prime } = 0 \]

6534

\[ {} \sqrt {a^{2}+x^{2}}\, \left (b {y^{\prime }}^{2}+y y^{\prime \prime }\right ) = y y^{\prime } \]

6535

\[ {} \sqrt {a^{2}-x^{2}}\, \left (-y y^{\prime }-x {y^{\prime }}^{2}+x y y^{\prime \prime }\right ) = b x {y^{\prime }}^{2} \]

6536

\[ {} \operatorname {f3} \left (x \right ) y^{2}+\operatorname {f2} \left (x \right ) y y^{\prime }+\operatorname {f1} \left (x \right ) {y^{\prime }}^{2}+\operatorname {f0} \left (x \right ) y y^{\prime \prime } = 0 \]

6537

\[ {} 4 f \left (x \right ) y y^{\prime \prime } = 4 f \left (x \right )^{2} y+3 f \left (x \right ) g \left (x \right ) y^{2}-f \left (x \right ) y^{4}+2 y^{3} f^{\prime }\left (x \right )+\left (-6 f \left (x \right ) y^{2}+2 f^{\prime }\left (x \right )\right ) y^{\prime }+3 f \left (x \right ) {y^{\prime }}^{2} \]

6541

\[ {} \left (1-2 y\right ) {y^{\prime }}^{2}+\left (1+y^{2}\right ) y^{\prime \prime } = 0 \]

6542

\[ {} \left (1+y^{2}\right ) y^{\prime \prime } = 3 y {y^{\prime }}^{2} \]

6543

\[ {} \left (1+y^{2}\right ) y^{\prime \prime } = \left (a +3 y\right ) {y^{\prime }}^{2} \]

6544

\[ {} y^{\prime } \left (1+{y^{\prime }}^{2}\right )+\left (1+y^{2}\right ) y^{\prime \prime } = 0 \]

6546

\[ {} \left (x +y^{2}\right ) y^{\prime \prime } = 2 \left (x -y^{2}\right ) {y^{\prime }}^{3}-y^{\prime } \left (1+4 y y^{\prime }\right ) \]

6547

\[ {} \left (x^{2}+y^{2}\right ) y^{\prime \prime } = \left (1+y^{2}\right ) \left (x y^{\prime }-y\right ) \]

6548

\[ {} \left (x^{2}+y^{2}\right ) y^{\prime \prime } = 2 \left (1+y^{2}\right ) \left (x y^{\prime }-y\right ) \]

6549

\[ {} 2 \left (1-y\right ) y y^{\prime \prime } = \left (1-2 y\right ) {y^{\prime }}^{2} \]

6550

\[ {} 2 \left (1-y\right ) y y^{\prime \prime } = f \left (x \right ) \left (1-y\right ) y y^{\prime }+\left (1-2 y\right ) {y^{\prime }}^{2} \]

6551

\[ {} 2 \left (1-y\right ) y y^{\prime \prime } = \left (1-3 y\right ) {y^{\prime }}^{2} \]

6552

\[ {} 2 \left (1-y\right ) y y^{\prime \prime } = 4 y \left (f \left (x \right )+g \left (x \right ) y\right ) y^{\prime }+\left (1-3 y\right ) {y^{\prime }}^{2} \]

6553

\[ {} 2 \left (1-y\right ) y y^{\prime \prime } = -\left (1-y\right )^{3} \left (\operatorname {F0} \left (x \right )^{2}-\operatorname {G0} \left (x \right )^{2} y^{2}\right )-4 \left (1-y\right ) y^{2} \left (f \left (x \right )^{2}-g \left (x \right )^{2}+f^{\prime }\left (x \right )+g^{\prime }\left (x \right )\right )-4 y \left (f \left (x \right )+g \left (x \right ) y\right ) y^{\prime }+\left (1-3 y\right ) {y^{\prime }}^{2} \]

6554

\[ {} 3 \left (1-y\right ) y y^{\prime \prime } = 2 \left (1-2 y\right ) {y^{\prime }}^{2} \]

6555

\[ {} 4 \left (1-y\right ) y y^{\prime \prime } = 3 \left (1-2 y\right ) {y^{\prime }}^{2} \]

6557

\[ {} x y^{2} y^{\prime \prime } = \left (a -y^{2}\right ) y^{\prime }+x y {y^{\prime }}^{2} \]

6558

\[ {} x^{2} y^{2} y^{\prime \prime } = \left (x^{2}+y^{2}\right ) \left (x y^{\prime }-y\right ) \]

6559

\[ {} \left (a^{2}-x^{2}\right ) y {y^{\prime }}^{2}+\left (a^{2}-x^{2}\right ) \left (a^{2}-y^{2}\right ) y^{\prime \prime } = x \left (a^{2}-y^{2}\right ) y^{\prime } \]

6560

\[ {} \operatorname {a2} x \left (1-y\right ) y^{2}+\operatorname {a3} \,x^{3} y^{2} \left (1+y\right )+\left (1-y\right )^{3} \left (\operatorname {a0} +\operatorname {a1} y^{2}\right )+2 x \left (1-y\right ) y y^{\prime }-x^{2} \left (1-3 y\right ) {y^{\prime }}^{2}+2 x^{2} \left (1-y\right ) y y^{\prime \prime } = 0 \]

6561

\[ {} \left (x +y\right ) \left (x y^{\prime }-y\right )^{3}+x^{3} y^{2} y^{\prime \prime } = 0 \]

6563

\[ {} \left (1-3 y^{2}\right ) {y^{\prime }}^{2}+y \left (1+y^{2}\right ) y^{\prime \prime } = 0 \]

6565

\[ {} \left (-\left (1-y\right ) \left (a -y\right )+y \left (1-y\right )+\left (a -y\right ) y\right ) {y^{\prime }}^{2}+2 \left (1-y\right ) \left (a -y\right ) y y^{\prime \prime } = \operatorname {a3} \left (1-y\right )^{2} \left (a -y\right )^{2}+\operatorname {a1} \left (1-y\right )^{2} y^{2}+\operatorname {a2} \left (a -y\right )^{2} y^{2}+\operatorname {a0} \left (a -y\right )^{2} y^{2} \left (1-y^{2}\right ) \]

6566

\[ {} \left (\left (a -y\right ) \left (b -y\right )+\left (a -y\right ) \left (c -y\right )+\left (b -y\right ) \left (c -y\right )\right ) {y^{\prime }}^{2}+2 \left (a -y\right ) \left (b -y\right ) \left (c -y\right ) y^{\prime \prime } = \operatorname {a3} \left (a -y\right )^{2} \left (b -y\right )^{2}+2 \operatorname {a2} \left (a -y\right )^{2} \left (c -y\right )^{2}+\operatorname {a1} \left (b -y\right )^{2} \left (c -y\right )^{2}+\operatorname {a0} \left (a -y\right )^{2} \left (b -y\right )^{2} \left (c -y\right )^{2} \]

6567

\[ {} 2 \left (1-x \right ) x \left (1-y\right ) \left (x -y\right ) y y^{\prime \prime } = -y^{2} \left (1-y^{2}\right )+2 \left (1-y\right ) y \left (x^{2}+y-2 x y\right ) y^{\prime }+\left (1-x \right ) x \left (x -2 y-2 x y+3 y^{2}\right ) {y^{\prime }}^{2} \]

6568

\[ {} 2 \left (1-x \right ) x \left (1-y\right ) \left (x -y\right ) y y^{\prime \prime } = f \left (x \right ) \left (\left (1-y\right ) \left (x -y\right ) y\right )^{{3}/{2}}-y^{2} \left (1-y^{2}\right )+2 \left (1-y\right ) y \left (x^{2}+y-2 x y\right ) y^{\prime }+\left (1-x \right ) x \left (x -2 y-2 x y+3 y^{2}\right ) {y^{\prime }}^{2} \]

6569

\[ {} 2 \left (1-x \right )^{2} x^{2} \left (1-y\right ) \left (x -y\right ) y y^{\prime \prime } = \operatorname {a0} x \left (1-y\right )^{2} \left (x -y\right )^{2}+\left (\operatorname {a2} -1\right ) \left (1-x \right ) x \left (1-y\right )^{2} y^{2}+\operatorname {a1} \left (1-x \right ) \left (x -y\right )^{2} y^{2}+\operatorname {a3} \left (1-y\right )^{2} \left (x -y\right )^{2} y^{2}+2 \left (1-x \right ) x \left (1-y\right )^{2} y \left (x^{2}+y-2 x y\right ) y^{\prime }+\left (1-x \right )^{2} x^{2} \left (x -2 y-2 x y+3 y^{2}\right ) {y^{\prime }}^{2} \]

6570

\[ {} y \left (1+a^{2}-2 a^{2} y^{2}\right )+b \sqrt {\left (1-y^{2}\right ) \left (1-a^{2} y^{2}\right )}\, {y^{\prime }}^{2}+\left (1-y^{2}\right ) \left (1-a^{2} y^{2}\right ) y^{\prime \prime } = 0 \]

6571

\[ {} a^{2} y+\left (x^{2}+y^{2}\right )^{2} y^{\prime \prime } = 0 \]

6572

\[ {} A y+\left (a +2 b x +c \,x^{2}+y^{2}\right )^{2} y^{\prime \prime } = 0 \]

6573

\[ {} \operatorname {f3} \left (y\right )+\operatorname {f2} \left (y\right ) y^{\prime }+\operatorname {f1} \left (y\right ) {y^{\prime }}^{2}+\operatorname {f0} \left (y\right ) y^{\prime \prime } = 0 \]

6577

\[ {} \operatorname {a2} \left (\operatorname {a3} +\operatorname {a1} \sin \left (y\right )^{2}\right ) y+\operatorname {a1} {y^{\prime }}^{2}+\operatorname {a1} \cos \left (y\right ) \sin \left (y\right ) {y^{\prime }}^{2}+\left (\operatorname {a0} +\operatorname {a1} \sin \left (y\right )^{2}\right ) y^{\prime \prime } = 0 \]

6578

\[ {} \left (1+\ln \left (y\right )\right ) {y^{\prime }}^{2}+\left (1-\ln \left (y\right )\right ) y y^{\prime \prime } = 0 \]

6580

\[ {} y^{\prime } y^{\prime \prime } = x y^{2}+x^{2} y y^{\prime } \]

6581

\[ {} y+x y^{\prime }+2 \left (x +y\right ) {y^{\prime }}^{2}+\left (y^{2}+2 x^{2} y^{\prime }\right ) y^{\prime \prime } = 0 \]

6582

\[ {} a y^{2}+x^{3} y^{\prime } y^{\prime \prime } = 0 \]

6583

\[ {} \operatorname {f5} y^{2}+\operatorname {f4} y y^{\prime }+\operatorname {f3} {y^{\prime }}^{2}+\operatorname {f2} y y^{\prime \prime }+\operatorname {f1} y^{\prime } y^{\prime \prime } = 0 \]

6585

\[ {} y+3 x y^{\prime }+2 {y^{\prime }}^{3} y+\left (x^{2}+2 y^{2} y^{\prime }\right ) y^{\prime \prime } = 0 \]

6587

\[ {} y^{3}+\left ({y^{\prime }}^{2}+y^{2}\right ) y^{\prime \prime } = 0 \]

6593

\[ {} y^{\prime }-x y^{\prime \prime }+{y^{\prime \prime }}^{2} = 0 \]

6594

\[ {} a^{2} {y^{\prime \prime }}^{2} = \left (1+{y^{\prime }}^{2}\right )^{3} \]

6597

\[ {} 2 \left (x -y^{\prime }\right ) y^{\prime }-x \left (x +4 y^{\prime }\right ) y^{\prime \prime }+2 \left (x^{2}+1\right ) {y^{\prime \prime }}^{2} = 2 y \]

6598

\[ {} 4 {y^{\prime }}^{2}-2 \left (3 x y^{\prime }+y\right ) y^{\prime \prime }+3 x^{2} {y^{\prime \prime }}^{2} = 0 \]

6599

\[ {} 6 y y^{\prime \prime }-6 \left (1-6 x \right ) x y^{\prime } y^{\prime \prime }+\left (2-9 x \right ) x^{2} {y^{\prime \prime }}^{2} = 36 x {y^{\prime }}^{2} \]

6600

\[ {} h y^{2}+\operatorname {g1} y y^{\prime }+\operatorname {g0} {y^{\prime }}^{2}+\operatorname {f2} y y^{\prime \prime }+\operatorname {f1} y^{\prime } y^{\prime \prime }+\operatorname {f0} {y^{\prime \prime }}^{2} = 0 \]

6601

\[ {} -{y^{\prime }}^{2}+4 {y^{\prime }}^{3} y+y y^{\prime \prime } = 0 \]

6602

\[ {} \left (1+{y^{\prime }}^{2}+y y^{\prime \prime }\right )^{2} = \left (1+{y^{\prime }}^{2}\right )^{3} \]

6603

\[ {} {y^{\prime }}^{2} \left (1-b^{2} {y^{\prime }}^{2}\right )+2 b^{2} y {y^{\prime }}^{2} y^{\prime \prime }+\left (a^{2}-b^{2} y^{2}\right ) {y^{\prime \prime }}^{2} = 0 \]

6604

\[ {} \left (y^{2}-x^{2} {y^{\prime }}^{2}+x^{2} y y^{\prime \prime }\right )^{2} = 4 x y \left (x y^{\prime }-y\right )^{3} \]

6605

\[ {} {y^{\prime \prime }}^{3} = 12 y^{\prime } \left (x y^{\prime \prime }-2 y^{\prime }\right ) \]