3.1.3 Problems 201 to 300

Table 3.5: First order ode

#

ODE

Mathematica

Maple

491

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

492

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

493

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

494

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

495

\[ {}y^{\prime } = \frac {-{\mathrm e}^{x}+3 x^{2}}{-5+2 y} \]

496

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

497

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

498

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

499

\[ {}y^{\prime } = \frac {3 x^{2}+1}{-6 y+3 y^{2}} \]

500

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

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

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

505

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

506

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

507

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

508

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

509

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

510

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

511

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

512

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

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

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

517

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

518

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

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

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

522

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

523

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

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

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

534

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

535

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

536

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

537

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

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

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

550

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

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

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

553

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

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

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

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

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

563

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

564

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

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

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

568

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

569

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

570

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

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

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

576

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

577

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

578

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

579

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

580

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

581

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

582

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

583

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

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

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