2.6 Problems 501 to 600

Table 2.6: Main lookup table

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ODE

Mathematica result

Maple result

501

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

502

\[ {}y^{\prime } = \frac {2-{\mathrm e}^{x}}{3+2 y} \]

503

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

504

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

505

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

506

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

507

\[ {}y^{\prime } = \frac {b +a y}{d +c y} \]

508

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

509

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

510

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

511

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

512

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

513

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

514

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

515

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

516

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

517

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

518

\[ {}\tan \relax (t ) y+y^{\prime } = \sin \relax (t ) \]

519

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

520

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

521

\[ {}y+\ln \relax (t ) y^{\prime } = \cot \relax (t ) \]

522

\[ {}y^{\prime } = \frac {t^{2}+1}{3 y-y^{2}} \]

523

\[ {}y^{\prime } = \frac {\cot \relax (t ) y}{1+y} \]

524

\[ {}y^{\prime } = -\frac {4 t}{y} \]

525

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

526

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

527

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

528

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

529

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

530

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

531

\[ {}y^{\prime } = t -1-y^{2} \]

532

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

533

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

534

\[ {}y^{\prime } = -1+{\mathrm e}^{y} \]

535

\[ {}y^{\prime } = -1+{\mathrm e}^{-y} \]

536

\[ {}y^{\prime } = -\frac {2 \arctan \relax (y)}{1+y^{2}} \]

537

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

538

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

539

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

540

\[ {}y^{\prime } = -b \sqrt {y}+a y \]

541

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

542

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

543

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

544

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

545

\[ {}2+3 x^{2}-2 x y+\left (3-x^{2}+6 y^{2}\right ) y^{\prime } = 0 \]

546

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

547

\[ {}y^{\prime } = \frac {-a x -b y}{b x +c y} \]

548

\[ {}y^{\prime } = \frac {-a x +b y}{b x -c y} \]

549

\[ {}{\mathrm e}^{x} \sin \relax (y)-2 \sin \relax (x ) y+\left (2 \cos \relax (x )+{\mathrm e}^{x} \cos \relax (y)\right ) y^{\prime } = 0 \]

550

\[ {}{\mathrm e}^{x} \sin \relax (y)+3 y-\left (3 x -{\mathrm e}^{x} \sin \relax (y)\right ) y^{\prime } = 0 \]

551

\[ {}2 x -2 \,{\mathrm e}^{x y} \sin \left (2 x \right )+{\mathrm e}^{x y} \cos \left (2 x \right ) y+\left (-3+{\mathrm e}^{x y} x \cos \left (2 x \right )\right ) y^{\prime } = 0 \]

552

\[ {}\frac {y}{x}+6 x +\left (\ln \relax (x )-2\right ) y^{\prime } = 0 \]

553

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

554

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

555

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

556

\[ {}-1+9 x^{2}+y+\left (x -4 y\right ) y^{\prime } = 0 \]

557

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

558

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

559

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

560

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

561

\[ {}y^{\prime } = -1+{\mathrm e}^{2 x}+y \]

562

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

563

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

564

\[ {}{\mathrm e}^{x}+\left ({\mathrm e}^{x} \cot \relax (y)+2 \csc \relax (y) y\right ) y^{\prime } = 0 \]

565

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

566

\[ {}3 x +\frac {6}{y}+\left (\frac {x^{2}}{y}+\frac {3 y}{x}\right ) y^{\prime } = 0 \]

567

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

568

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

569

\[ {}y^{\prime } = \frac {\cos \relax (x )+1}{2-\sin \relax (y)} \]

570

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

571

\[ {}y^{\prime } = 3-6 x +y-2 x y \]

572

\[ {}y^{\prime } = \frac {-1-2 x y-y^{2}}{x^{2}+2 x y} \]

573

\[ {}x y+x y^{\prime } = 1-y \]

574

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

575

\[ {}2 y+x y^{\prime } = \frac {\sin \relax (x )}{x} \]

576

\[ {}y^{\prime } = \frac {-1-2 x y}{x^{2}+2 y} \]

577

\[ {}\frac {-x^{2}+x +1}{x^{2}}+\frac {y y^{\prime }}{y-2} = 0 \]

578

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

579

\[ {}y+y^{\prime } = \frac {1}{1+{\mathrm e}^{x}} \]

580

\[ {}y^{\prime } = 1+2 x +y^{2}+2 x y^{2} \]

581

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

582

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

583

\[ {}y^{\prime } = \frac {-{\mathrm e}^{2 y} \cos \relax (x )+\cos \relax (y) {\mathrm e}^{-x}}{2 \,{\mathrm e}^{2 y} \sin \relax (x )-\sin \relax (y) {\mathrm e}^{-x}} \]

584

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

585

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

586

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

587

\[ {}y^{\prime } = {\mathrm e}^{x +y} \]

588

\[ {}\frac {-4+6 x y+2 y^{2}}{3 x^{2}+4 x y+3 y^{2}}+y^{\prime } = 0 \]

589

\[ {}y^{\prime } = \frac {x^{2}-1}{1+y^{2}} \]

590

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

591

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

592

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

593

\[ {}x y^{\prime } = {\mathrm e}^{\frac {y}{x}} x +y \]

594

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

595

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

596

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

597

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

598

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

599

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

600

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