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1.3E: Direction Fields (Exercises)

  • Page ID
    103466
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    In Exercises 1–10 a direction field is drawn for the given equation. Sketch a solution for the IVP.

    1. \(y'= {\frac{x}{y}}, \quad y(0)=1\)
    Screenshot (17).png
    Figure 1.3E.1 : A direction field for \(y'= {\frac{x}{y}}\)
    2. \( {y'= {2xy^2\over1+x^2}} \quad y(1)=-1\)
    Screenshot (20).png
    Figure 1.3E.2 : A direction field for \( {y'= {2xy^2\over1+x^2}}\)
    3. \(y'=x^2(1+y^2) \quad y(0)=0\)
    clipboard_e69301f6df8e696be899ba6bc0a6b8680.png
    Figure 1.3E.3 : A direction field for \(y'=x^2(1+y^2)\)
    4. \(y'= {1\over1+x^2+y^2} \quad y(0)=-1\)
    clipboard_e4f5bd8677d21973d252773974df4bb5a.png
    Figure 1.3E.4 : A direction field for \(y'= {1\over1+x^2+y^2}\)
    5. \(y'=-(2xy^2+y^3) \quad y(-1)=1\)
    clipboard_e1e89fb6a79530c9a62d3605e9f6665ee.png
    Figure 1.3E.5 : A direction field for \(y'=-(2xy^2+y^3)\)
    6. \(y'=(x^2+y^2)^{1/2} \quad y(0)=0\)
    clipboard_ef4c64f8be3b1000f5fc1873f6555d847.png
    Figure 1.3E.6 : A direction field for \(y'=(x^2+y^2)^{1/2}\)
    7. \(y'=\sin xy \quad y(1)=-1\)
    clipboard_e6ac2066545ff34a67f082137261cf496.png
    Figure 1.3E.7 : A direction field for \(y'=\sin xy\)
    8. \(y'=e^{xy} \quad y(0)=1\)
    clipboard_eaa889c93e25125dddb2aafbd727527ee.png
    Figure 1.3E.8 : A direction field for \(y'=e^{xy}\)
    9. \(y'=x^3y^2+xy^3 \quad y(0)=0\)
    clipboard_ed44ee0c5ae75c2e192d5759eb93ae2d8.png
    Figure 1.3E.9 : A direction field for \(y'=x^3y^2+xy^3\)
    10. \(y'=\sin(x-2y) \quad y(0)=0\)
    clipboard_ea368b9789d8a64aa2f105a7deb526449.png
    Figure 1.3E.10 : A direction field for \(y'=\sin(x-2y)\)

    In Exercises 11 - 20 use the slope field generator to construct a solution curve to the given IVP in the indicated rectangular region.

    11. \(y'=y(y-1), \quad y(1)=1; \quad \{-1\le x\le 2,\ -2\le y\le2\}\)

    12. \(y'=2-3xy, \quad y(0)=0; \quad \{-2\le x\le 2,\ -2\le y\le2\}\)

    13. \(y'=xy(y-1), \quad y(0)=1; \quad \{-2\le x\le2,\ -3\le y\le 4\}\)

    14. \(y'=3x+y, \quad y(0)=1; \quad \{-2\le x\le2,\ 0\le y\le 4\}\)

    15. \(y'=y-x^3, \quad y(1)=1; \quad \{-2\le x\le2,\ -2\le y\le 2\}\)

    16. \(y'=1-x^2-y^2, \quad y(0)=0; \quad \{-2\le x\le2,\ -2\le y\le 2\}\)

    17. \(y'=x(y^2-1), \quad y(1)=2; \quad \{-3\le x\le3,\ -3\le y\le 3\}\)

    18. \(y'= {xy^2\over y-1}, \quad y(0)=3; \quad \{-1\le x\le1,\ 1\le y\le 4\}\)

    19. \(y'=\sin y, \quad y(1)=1; \quad \{-2\le x\le2,\ -2\le y\le 2\}\)

    20. \(y'={x^2+y^2\over1+x^2+y^2}, \quad y(1)=0; \quad \{-2\le x\le2,\ -2\le y\le 2\}\)


    This page titled 1.3E: Direction Fields (Exercises) is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench via source content that was edited to the style and standards of the LibreTexts platform.