{"id":181,"date":"2026-01-12T16:00:48","date_gmt":"2026-01-12T16:00:48","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/?post_type=chapter&p=181"},"modified":"2026-01-12T16:00:49","modified_gmt":"2026-01-12T16:00:49","slug":"introduction-to-differential-equations-get-stronger-answer-key","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/chapter\/introduction-to-differential-equations-get-stronger-answer-key\/","title":{"raw":"Introduction to Differential Equations: Get Stronger Answer Key","rendered":"Introduction to Differential Equations: Get Stronger Answer Key"},"content":{"raw":"

Basics of Differential Equations<\/span><\/h2>\r\n
    \r\n \t
  1. [latex]1[\/latex]<\/li>\r\n \t
  2. [latex]3[\/latex]<\/li>\r\n \t
  3. [latex]1[\/latex]<\/li>\r\n \t
  4. [latex]1[\/latex]<\/li>\r\n \t
  5. <\/li>\r\n \t
  6. <\/li>\r\n \t
  7. <\/li>\r\n \t
  8. <\/li>\r\n \t
  9. <\/li>\r\n \t
  10. [latex]y=4+\\dfrac{3{x}^{4}}{4}[\/latex]<\/li>\r\n \t
  11. [latex]y=\\dfrac{1}{2}{e}^{{x}^{2}}[\/latex]<\/li>\r\n \t
  12. [latex]y=2{e}^{\\dfrac{\\text{-}1}{x}}[\/latex]<\/li>\r\n \t
  13. [latex]u={\\sin}^{-1}\\left({e}^{-1+t}\\right)[\/latex]<\/li>\r\n \t
  14. [latex]y=-\\dfrac{\\sqrt{x+1}}{\\sqrt{1-x}}-1[\/latex]<\/li>\r\n \t
  15. [latex]y=C-x+x\\text{ln}x-\\text{ln}\\left(\\cos{x}\\right)[\/latex]<\/li>\r\n \t
  16. [latex]y=C+\\dfrac{{4}^{x}}{\\text{ln}\\left(4\\right)}[\/latex]<\/li>\r\n \t
  17. [latex]y=\\dfrac{2}{3}\\sqrt{{t}^{2}+16}\\left({t}^{2}+16\\right)+C[\/latex]<\/li>\r\n \t
  18. [latex]x=\\dfrac{2}{15}\\sqrt{4+t}\\left(3{t}^{2}+4t - 32\\right)+C[\/latex]<\/li>\r\n \t
  19. [latex]y=Cx[\/latex]<\/li>\r\n \t
  20. [latex]y=1-\\dfrac{{t}^{2}}{2},y=-\\dfrac{{t}^{2}}{2}-1[\/latex]<\/li>\r\n \t
  21. [latex]y={e}^{\\text{-}t},y=\\text{-}{e}^{\\text{-}t}[\/latex]<\/li>\r\n \t
  22. [latex]y=2\\left({t}^{2}+5\\right),t=3\\sqrt{5}[\/latex]<\/li>\r\n \t
  23. [latex]y=10{e}^{-2t},t=-\\dfrac{1}{2}\\text{ln}\\left(\\dfrac{1}{10}\\right)[\/latex]<\/li>\r\n \t
  24. [latex]y=\\dfrac{1}{4}\\left(41-{e}^{-4t}\\right)[\/latex], never<\/li>\r\n \t
  25. Solution changes from increasing to decreasing at [latex]y\\left(0\\right)=0[\/latex]<\/li>\r\n \t
  26. Solution changes from increasing to decreasing at [latex]y\\left(0\\right)=0[\/latex]<\/li>\r\n \t
  27. [latex]v\\left(t\\right)=-32t+a[\/latex]<\/li>\r\n \t
  28. [latex]0[\/latex] ft\/s<\/li>\r\n \t
  29. [latex]y=4{e}^{3t}[\/latex]<\/li>\r\n \t
  30. [latex]y=3 - 2t+{t}^{2}[\/latex]<\/li>\r\n \t
  31. [latex]y=\\dfrac{1}{k}\\left({e}^{kt}-1\\right)[\/latex] and [latex]y=x[\/latex]<\/li>\r\n<\/ol>\r\n

    Direction Fields and Euler's Method<\/span><\/h2>\r\n
      \r\n \t
    1. \"A<\/span><\/li>\r\n \t
    2. [latex]y=0[\/latex] is a stable equilibrium<\/li>\r\n \t
    3. \"A<\/li>\r\n \t
    4. [latex]y=0[\/latex] is a stable equilibrium and [latex]y=2[\/latex] is unstable<\/li>\r\n \t
    5. \"A<\/li>\r\n \t
    6. \"A<\/li>\r\n \t
    7. \"A<\/span><\/li>\r\n \t
    8. \"A<\/li>\r\n \t
    9. \"A<\/li>\r\n \t
    10. E<\/li>\r\n \t
    11. A<\/li>\r\n \t
    12. B<\/li>\r\n \t
    13. A<\/li>\r\n \t
    14. C<\/li>\r\n \t
    15. [latex]2.24[\/latex], exact: [latex]3[\/latex]<\/li>\r\n \t
    16. [latex]7.739364[\/latex], exact: [latex]5\\left(e - 1\\right)[\/latex]<\/li>\r\n \t
    17. [latex]-0.2535[\/latex], exact: [latex]0[\/latex]<\/li>\r\n \t
    18. [latex]1.345[\/latex], exact: [latex]\\dfrac{1}{\\text{ln}\\left(2\\right)}[\/latex]<\/li>\r\n \t
    19. [latex]-4[\/latex], exact: [latex]\\dfrac{\\text{-}1}{2}[\/latex]<\/li>\r\n \t
    20. \"A<\/li>\r\n \t
    21. [latex]4.0741{e}^{-10}[\/latex]<\/li>\r\n<\/ol>\r\n

      Separation of Variables<\/span><\/h2>\r\n
        \r\n \t
      1. [latex]y={e}^{t}-1[\/latex]<\/li>\r\n \t
      2. [latex]y=1-{e}^{\\text{-}t}[\/latex]<\/li>\r\n \t
      3. [latex]y=Cx{e}^{\\dfrac{-1}{x}}[\/latex]<\/li>\r\n \t
      4. [latex]y=\\dfrac{1}{C-{x}^{2}}[\/latex]<\/li>\r\n \t
      5. [latex]y=-\\dfrac{2}{C+\\text{ln}x}[\/latex]<\/li>\r\n \t
      6. [latex]y=C{e}^{x}\\left(x+1\\right)+1[\/latex]<\/li>\r\n \t
      7. [latex]y=\\sin\\left(\\text{ln}t+C\\right)[\/latex]<\/li>\r\n \t
      8. [latex]y=\\text{-}\\text{ln}\\left({e}^{\\text{-}x}\\right)[\/latex]<\/li>\r\n \t
      9. [latex]y=\\dfrac{1}{\\sqrt{2-{e}^{{x}^{2}}}}[\/latex]<\/li>\r\n \t
      10. [latex]y={\\text{tanh}}^{-1}\\left(\\dfrac{{x}^{2}}{2}\\right)[\/latex]<\/li>\r\n \t
      11. [latex]x=\\sin\\left(1-t+t\\text{ln}t\\right)[\/latex]<\/li>\r\n \t
      12. [latex]y=\\text{ln}\\left(\\text{ln}\\left(5\\right)\\right)-\\text{ln}\\left(2-{5}^{x}\\right)[\/latex]<\/li>\r\n \t
      13. [latex]y=C{e}^{-2x}+\\dfrac{1}{2}[\/latex]<\/li>\r\n \t
      14. [latex]y=\\dfrac{1}{\\sqrt{2}\\sqrt{C-{e}^{x}}}[\/latex]<\/li>\r\n \t
      15. [latex]y=C{e}^{\\text{-}x}{x}^{x}[\/latex]<\/li>\r\n \t
      16. [latex]y=\\dfrac{r}{d}\\left(1-{e}^{\\text{-}dt}\\right)[\/latex]<\/li>\r\n \t
      17. [latex]y\\left(t\\right)=10 - 9{e}^{\\dfrac{\\text{-}x}{50}}[\/latex]<\/li>\r\n \t
      18. [latex]T\\left(t\\right)=20+50{e}^{-0.125t}[\/latex]<\/li>\r\n \t
      19. [latex]T\\left(t\\right)=20+38.5{e}^{-0.125t}[\/latex]<\/li>\r\n \t
      20. [latex]24[\/latex] hours [latex]57[\/latex] minutes<\/li>\r\n \t
      21. [latex]T\\left(t\\right)=20+50{e}^{-0.125t}[\/latex]<\/li>\r\n \t
      22. [latex]T\\left(t\\right)=20+38.5{e}^{-0.125t}[\/latex]<\/li>\r\n \t
      23. [latex]y=\\left(c+\\dfrac{b}{a}\\right){e}^{ax}-\\dfrac{b}{a}[\/latex]<\/li>\r\n \t
      24. [latex]y\\left(t\\right)=cL+\\left(I-cL\\right){e}^{\\dfrac{\\text{-}rt}{L}}[\/latex]<\/li>\r\n \t
      25. [latex]y=40\\left(1-{e}^{-0.1t}\\right),40[\/latex] g\/cm\u00b2<\/li>\r\n<\/ol>\r\n

        First-Order Linear Equations and Applications<\/span><\/h2>\r\n
          \r\n \t
        1. Yes<\/li>\r\n \t
        2. Yes<\/li>\r\n \t
        3. [latex]y^{\\prime} -{x}^{3}y=\\sin{x}[\/latex]<\/li>\r\n \t
        4. [latex]y^{\\prime} +\\dfrac{\\left(3x+2\\right)}{x}y=\\text{-}{e}^{x}[\/latex]<\/li>\r\n \t
        5. [latex]\\dfrac{dy}{dt}-yx\\left(x+1\\right)=0[\/latex]<\/li>\r\n \t
        6. [latex]{e}^{x}[\/latex]<\/li>\r\n \t
        7. [latex]\\text{-}\\text{ln}\\left(\\text{cosh}x\\right)[\/latex]<\/li>\r\n \t
        8. [latex]y=C{e}^{3x}-\\dfrac{2}{3}[\/latex]<\/li>\r\n \t
        9. [latex]y=C{x}^{3}+6{x}^{2}[\/latex]<\/li>\r\n \t
        10. [latex]y=C{e}^{\\dfrac{{x}^{2}}{2}}-3[\/latex]<\/li>\r\n \t
        11. [latex]y=C\\tan\\left(\\dfrac{x}{2}\\right)-2x+4\\tan\\left(\\dfrac{x}{2}\\right)\\text{ln}\\left(\\sin\\left(\\dfrac{x}{2}\\right)\\right)[\/latex]<\/li>\r\n \t
        12. [latex]y=C{x}^{3}-{x}^{2}[\/latex]<\/li>\r\n \t
        13. [latex]y=C{\\left(x+2\\right)}^{2}+\\dfrac{1}{2}[\/latex]<\/li>\r\n \t
        14. [latex]y=\\dfrac{C}{\\sqrt{x}}+2\\sin\\left(3t\\right)[\/latex]<\/li>\r\n \t
        15. [latex]y=C{\\left(x+1\\right)}^{3}-{x}^{2}-2x - 1[\/latex]<\/li>\r\n \t
        16. [latex]y=C{e}^{{\\text{sinh}}^{-1}x}-2[\/latex]<\/li>\r\n \t
        17. [latex]y=x+4{e}^{x}-1[\/latex]<\/li>\r\n \t
        18. [latex]y=-\\dfrac{3x}{2}\\left({x}^{2}-1\\right)[\/latex]<\/li>\r\n \t
        19. [latex]y=1-{e}^{{\\tan}^{-1}x}[\/latex]<\/li>\r\n \t
        20. [latex]y=\\left(x+2\\right)\\text{ln}\\left(\\dfrac{x+2}{2}\\right)[\/latex]<\/li>\r\n \t
        21. [latex]y=2{e}^{2\\sqrt{x}}-2x - 2\\sqrt{x}-1[\/latex]<\/li>\r\n \t
        22. [latex]v\\left(t\\right)=\\dfrac{gm}{k}\\left(1-{e}^{\\dfrac{\\text{-}kt}{m}}\\right)[\/latex]<\/li>\r\n \t
        23. [latex]\\sqrt{\\dfrac{gm}{k}}[\/latex]<\/li>\r\n \t
        24. [latex]40.451[\/latex] seconds<\/li>\r\n \t
        25. [latex]y=C{e}^{x}-a\\left(x+1\\right)[\/latex]<\/li>\r\n \t
        26. [latex]y=C{e}^{\\dfrac{{x}^{2}}{2}}-a[\/latex]<\/li>\r\n \t
        27. [latex]P=0[\/latex] semi-stable<\/li>\r\n \t
        28. [latex]P=\\dfrac{10{e}^{10x}}{{e}^{10x}+4}[\/latex]<\/li>\r\n \t
        29. [latex]P\\left(t\\right)=\\dfrac{10000{e}^{0.02t}}{150+50{e}^{0.02t}}[\/latex]<\/li>\r\n \t
        30. [latex]69[\/latex] hours [latex]5[\/latex] minutes<\/li>\r\n \t
        31. [latex]8[\/latex] years [latex]11[\/latex] months<\/li>\r\n \t
        32. [latex]y=\\dfrac{-20}{4\\times {10}^{-6}-0.002{e}^{0.01t}}[\/latex]<\/li>\r\n \t
        33. [latex]P\\left(t\\right)=\\dfrac{850+500{e}^{0.009t}}{85+5{e}^{0.009t}}[\/latex]<\/li>\r\n \t
        34. [latex]13[\/latex] years<\/li>\r\n \t
        35. [latex]31.465[\/latex] days<\/li>\r\n \t
        36. September [latex]2008[\/latex]<\/li>\r\n \t
        37. [latex]\\dfrac{K+T}{2}[\/latex]<\/li>\r\n \t
        38. [latex]r=0.0405[\/latex]<\/li>\r\n \t
        39. <\/li>\r\n \t
        40. [latex]\\alpha =0.0081[\/latex]<\/li>\r\n \t
        41. Logistic: [latex]361[\/latex], Threshold: [latex]436[\/latex], Gompertz: [latex]309[\/latex].<\/li>\r\n<\/ol>","rendered":"

          Basics of Differential Equations<\/span><\/h2>\n
            \n
          1. [latex]1[\/latex]<\/li>\n
          2. [latex]3[\/latex]<\/li>\n
          3. [latex]1[\/latex]<\/li>\n
          4. [latex]1[\/latex]<\/li>\n
          5. <\/li>\n
          6. <\/li>\n
          7. <\/li>\n
          8. <\/li>\n
          9. <\/li>\n
          10. [latex]y=4+\\dfrac{3{x}^{4}}{4}[\/latex]<\/li>\n
          11. [latex]y=\\dfrac{1}{2}{e}^{{x}^{2}}[\/latex]<\/li>\n
          12. [latex]y=2{e}^{\\dfrac{\\text{-}1}{x}}[\/latex]<\/li>\n
          13. [latex]u={\\sin}^{-1}\\left({e}^{-1+t}\\right)[\/latex]<\/li>\n
          14. [latex]y=-\\dfrac{\\sqrt{x+1}}{\\sqrt{1-x}}-1[\/latex]<\/li>\n
          15. [latex]y=C-x+x\\text{ln}x-\\text{ln}\\left(\\cos{x}\\right)[\/latex]<\/li>\n
          16. [latex]y=C+\\dfrac{{4}^{x}}{\\text{ln}\\left(4\\right)}[\/latex]<\/li>\n
          17. [latex]y=\\dfrac{2}{3}\\sqrt{{t}^{2}+16}\\left({t}^{2}+16\\right)+C[\/latex]<\/li>\n
          18. [latex]x=\\dfrac{2}{15}\\sqrt{4+t}\\left(3{t}^{2}+4t - 32\\right)+C[\/latex]<\/li>\n
          19. [latex]y=Cx[\/latex]<\/li>\n
          20. [latex]y=1-\\dfrac{{t}^{2}}{2},y=-\\dfrac{{t}^{2}}{2}-1[\/latex]<\/li>\n
          21. [latex]y={e}^{\\text{-}t},y=\\text{-}{e}^{\\text{-}t}[\/latex]<\/li>\n
          22. [latex]y=2\\left({t}^{2}+5\\right),t=3\\sqrt{5}[\/latex]<\/li>\n
          23. [latex]y=10{e}^{-2t},t=-\\dfrac{1}{2}\\text{ln}\\left(\\dfrac{1}{10}\\right)[\/latex]<\/li>\n
          24. [latex]y=\\dfrac{1}{4}\\left(41-{e}^{-4t}\\right)[\/latex], never<\/li>\n
          25. Solution changes from increasing to decreasing at [latex]y\\left(0\\right)=0[\/latex]<\/li>\n
          26. Solution changes from increasing to decreasing at [latex]y\\left(0\\right)=0[\/latex]<\/li>\n
          27. [latex]v\\left(t\\right)=-32t+a[\/latex]<\/li>\n
          28. [latex]0[\/latex] ft\/s<\/li>\n
          29. [latex]y=4{e}^{3t}[\/latex]<\/li>\n
          30. [latex]y=3 - 2t+{t}^{2}[\/latex]<\/li>\n
          31. [latex]y=\\dfrac{1}{k}\\left({e}^{kt}-1\\right)[\/latex] and [latex]y=x[\/latex]<\/li>\n<\/ol>\n

            Direction Fields and Euler’s Method<\/span><\/h2>\n
              \n
            1. \"A<\/span><\/li>\n
            2. [latex]y=0[\/latex] is a stable equilibrium<\/li>\n
            3. \"A<\/li>\n
            4. [latex]y=0[\/latex] is a stable equilibrium and [latex]y=2[\/latex] is unstable<\/li>\n
            5. \"A<\/li>\n
            6. \"A<\/li>\n
            7. \"A<\/span><\/li>\n
            8. \"A<\/li>\n
            9. \"A<\/li>\n
            10. E<\/li>\n
            11. A<\/li>\n
            12. B<\/li>\n
            13. A<\/li>\n
            14. C<\/li>\n
            15. [latex]2.24[\/latex], exact: [latex]3[\/latex]<\/li>\n
            16. [latex]7.739364[\/latex], exact: [latex]5\\left(e - 1\\right)[\/latex]<\/li>\n
            17. [latex]-0.2535[\/latex], exact: [latex]0[\/latex]<\/li>\n
            18. [latex]1.345[\/latex], exact: [latex]\\dfrac{1}{\\text{ln}\\left(2\\right)}[\/latex]<\/li>\n
            19. [latex]-4[\/latex], exact: [latex]\\dfrac{\\text{-}1}{2}[\/latex]<\/li>\n
            20. \"A<\/li>\n
            21. [latex]4.0741{e}^{-10}[\/latex]<\/li>\n<\/ol>\n

              Separation of Variables<\/span><\/h2>\n
                \n
              1. [latex]y={e}^{t}-1[\/latex]<\/li>\n
              2. [latex]y=1-{e}^{\\text{-}t}[\/latex]<\/li>\n
              3. [latex]y=Cx{e}^{\\dfrac{-1}{x}}[\/latex]<\/li>\n
              4. [latex]y=\\dfrac{1}{C-{x}^{2}}[\/latex]<\/li>\n
              5. [latex]y=-\\dfrac{2}{C+\\text{ln}x}[\/latex]<\/li>\n
              6. [latex]y=C{e}^{x}\\left(x+1\\right)+1[\/latex]<\/li>\n
              7. [latex]y=\\sin\\left(\\text{ln}t+C\\right)[\/latex]<\/li>\n
              8. [latex]y=\\text{-}\\text{ln}\\left({e}^{\\text{-}x}\\right)[\/latex]<\/li>\n
              9. [latex]y=\\dfrac{1}{\\sqrt{2-{e}^{{x}^{2}}}}[\/latex]<\/li>\n
              10. [latex]y={\\text{tanh}}^{-1}\\left(\\dfrac{{x}^{2}}{2}\\right)[\/latex]<\/li>\n
              11. [latex]x=\\sin\\left(1-t+t\\text{ln}t\\right)[\/latex]<\/li>\n
              12. [latex]y=\\text{ln}\\left(\\text{ln}\\left(5\\right)\\right)-\\text{ln}\\left(2-{5}^{x}\\right)[\/latex]<\/li>\n
              13. [latex]y=C{e}^{-2x}+\\dfrac{1}{2}[\/latex]<\/li>\n
              14. [latex]y=\\dfrac{1}{\\sqrt{2}\\sqrt{C-{e}^{x}}}[\/latex]<\/li>\n
              15. [latex]y=C{e}^{\\text{-}x}{x}^{x}[\/latex]<\/li>\n
              16. [latex]y=\\dfrac{r}{d}\\left(1-{e}^{\\text{-}dt}\\right)[\/latex]<\/li>\n
              17. [latex]y\\left(t\\right)=10 - 9{e}^{\\dfrac{\\text{-}x}{50}}[\/latex]<\/li>\n
              18. [latex]T\\left(t\\right)=20+50{e}^{-0.125t}[\/latex]<\/li>\n
              19. [latex]T\\left(t\\right)=20+38.5{e}^{-0.125t}[\/latex]<\/li>\n
              20. [latex]24[\/latex] hours [latex]57[\/latex] minutes<\/li>\n
              21. [latex]T\\left(t\\right)=20+50{e}^{-0.125t}[\/latex]<\/li>\n
              22. [latex]T\\left(t\\right)=20+38.5{e}^{-0.125t}[\/latex]<\/li>\n
              23. [latex]y=\\left(c+\\dfrac{b}{a}\\right){e}^{ax}-\\dfrac{b}{a}[\/latex]<\/li>\n
              24. [latex]y\\left(t\\right)=cL+\\left(I-cL\\right){e}^{\\dfrac{\\text{-}rt}{L}}[\/latex]<\/li>\n
              25. [latex]y=40\\left(1-{e}^{-0.1t}\\right),40[\/latex] g\/cm\u00b2<\/li>\n<\/ol>\n

                First-Order Linear Equations and Applications<\/span><\/h2>\n
                  \n
                1. Yes<\/li>\n
                2. Yes<\/li>\n
                3. [latex]y^{\\prime} -{x}^{3}y=\\sin{x}[\/latex]<\/li>\n
                4. [latex]y^{\\prime} +\\dfrac{\\left(3x+2\\right)}{x}y=\\text{-}{e}^{x}[\/latex]<\/li>\n
                5. [latex]\\dfrac{dy}{dt}-yx\\left(x+1\\right)=0[\/latex]<\/li>\n
                6. [latex]{e}^{x}[\/latex]<\/li>\n
                7. [latex]\\text{-}\\text{ln}\\left(\\text{cosh}x\\right)[\/latex]<\/li>\n
                8. [latex]y=C{e}^{3x}-\\dfrac{2}{3}[\/latex]<\/li>\n
                9. [latex]y=C{x}^{3}+6{x}^{2}[\/latex]<\/li>\n
                10. [latex]y=C{e}^{\\dfrac{{x}^{2}}{2}}-3[\/latex]<\/li>\n
                11. [latex]y=C\\tan\\left(\\dfrac{x}{2}\\right)-2x+4\\tan\\left(\\dfrac{x}{2}\\right)\\text{ln}\\left(\\sin\\left(\\dfrac{x}{2}\\right)\\right)[\/latex]<\/li>\n
                12. [latex]y=C{x}^{3}-{x}^{2}[\/latex]<\/li>\n
                13. [latex]y=C{\\left(x+2\\right)}^{2}+\\dfrac{1}{2}[\/latex]<\/li>\n
                14. [latex]y=\\dfrac{C}{\\sqrt{x}}+2\\sin\\left(3t\\right)[\/latex]<\/li>\n
                15. [latex]y=C{\\left(x+1\\right)}^{3}-{x}^{2}-2x - 1[\/latex]<\/li>\n
                16. [latex]y=C{e}^{{\\text{sinh}}^{-1}x}-2[\/latex]<\/li>\n
                17. [latex]y=x+4{e}^{x}-1[\/latex]<\/li>\n
                18. [latex]y=-\\dfrac{3x}{2}\\left({x}^{2}-1\\right)[\/latex]<\/li>\n
                19. [latex]y=1-{e}^{{\\tan}^{-1}x}[\/latex]<\/li>\n
                20. [latex]y=\\left(x+2\\right)\\text{ln}\\left(\\dfrac{x+2}{2}\\right)[\/latex]<\/li>\n
                21. [latex]y=2{e}^{2\\sqrt{x}}-2x - 2\\sqrt{x}-1[\/latex]<\/li>\n
                22. [latex]v\\left(t\\right)=\\dfrac{gm}{k}\\left(1-{e}^{\\dfrac{\\text{-}kt}{m}}\\right)[\/latex]<\/li>\n
                23. [latex]\\sqrt{\\dfrac{gm}{k}}[\/latex]<\/li>\n
                24. [latex]40.451[\/latex] seconds<\/li>\n
                25. [latex]y=C{e}^{x}-a\\left(x+1\\right)[\/latex]<\/li>\n
                26. [latex]y=C{e}^{\\dfrac{{x}^{2}}{2}}-a[\/latex]<\/li>\n
                27. [latex]P=0[\/latex] semi-stable<\/li>\n
                28. [latex]P=\\dfrac{10{e}^{10x}}{{e}^{10x}+4}[\/latex]<\/li>\n
                29. [latex]P\\left(t\\right)=\\dfrac{10000{e}^{0.02t}}{150+50{e}^{0.02t}}[\/latex]<\/li>\n
                30. [latex]69[\/latex] hours [latex]5[\/latex] minutes<\/li>\n
                31. [latex]8[\/latex] years [latex]11[\/latex] months<\/li>\n
                32. [latex]y=\\dfrac{-20}{4\\times {10}^{-6}-0.002{e}^{0.01t}}[\/latex]<\/li>\n
                33. [latex]P\\left(t\\right)=\\dfrac{850+500{e}^{0.009t}}{85+5{e}^{0.009t}}[\/latex]<\/li>\n
                34. [latex]13[\/latex] years<\/li>\n
                35. [latex]31.465[\/latex] days<\/li>\n
                36. September [latex]2008[\/latex]<\/li>\n
                37. [latex]\\dfrac{K+T}{2}[\/latex]<\/li>\n
                38. [latex]r=0.0405[\/latex]<\/li>\n
                39. <\/li>\n
                40. [latex]\\alpha =0.0081[\/latex]<\/li>\n
                41. Logistic: [latex]361[\/latex], Threshold: [latex]436[\/latex], Gompertz: [latex]309[\/latex].<\/li>\n<\/ol>\n","protected":false},"author":15,"menu_order":8,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":109,"module-header":"- Select Header -","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/181"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":1,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/181\/revisions"}],"predecessor-version":[{"id":193,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/181\/revisions\/193"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/parts\/109"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/181\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/media?parent=181"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapter-type?post=181"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/contributor?post=181"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/license?post=181"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}