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Mathematics LibreTexts

3.9.E: Exercises

  • Page ID
    219719
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    3.9 Exercises

    Exercise \(\PageIndex{1}-\PageIndex{4}\)

    In problems 1 – 4, check that the function \(y\) is a solution of the given differential equation.

    1. \(y' + 3y = 6\). \(y = e^{–3x} + 2\). 2. \(y' – 2y = 8\). \(y = e^{2x} – 4\).
    3. \(y' = – x/y\). \(y = \sqrt{7-x^2}\). 4. \(y' = x – y\). \(y = x – 1 + 2e^{–x}\).
    Exercise \(\PageIndex{5}-\PageIndex{8}\)

    In problems 5 – 8 check that the function \(y\) is a solution of the given initial value problem.

    5. \(y' = 6x^2 – 3\) and \(y(1) = 2 \). \(y = 2x^3 – 3x + 3\).

    6. \(y' = 6x + 4\) and \(y(2) = 3\). \(y = 3x^2 + 4x – 17\).

    7. \(y' = 5y\) and \(y(0) = 7\). \(y = 7e^{5x}\).

    8. \(y' = –2y\) and \(y(0) = 3\). \(y = 3e^{–2x}\).

    Exercise \(\PageIndex{9}-\PageIndex{12}\)

    In problems 9 – 12, a family of solutions of a differential equation is given. Find the value of the constant \(C\) so the solution satisfies the initial value condition.

    9. \(y' = 2x\) and \(y(3) = 7\). \(y = x^2 + C\). 10. \(y' = 3x^2 – 5\) and \(y(1) = 2\). \(y = x^3 – 5x + C\).
    11. \(y' = 3y\) and \(y(0) = 5\). \(y = Ce^{3x}\). 12. \(y' = –2y\) and \(y(0) = 3\). \(y = Ce^{–2x}\).
    Exercise \(\PageIndex{13}-\PageIndex{18}\)

    In problems 13 – 18, solve the differential equation. (Assume that \(x\) and \(y\) are restricted so that division by zero does not occur.)

    13. \(y' = 2xy\) 14. \(y' = x/y\) 15. \(xy' = y + 3\)
    16. \(y' = x^2y + 3y\) 17. \(y' = 4y\) 18. \(y' = 5(2 – y)\)
    Exercise \(\PageIndex{19}-\PageIndex{22}\)

    In problems 19 – 22, solve the initial value separable differential equations.

    19. \(y' = 2xy\) for \(y(0) = 3\), \(y(0) = 5\), and \(y(1) = 2\).

    20. \(y' = x/y\) for \(y(0) = 3\), \(y(0) = 5\), and \(y(1) = 2\).

    21. \(y' = 3y\) for \(y(0) = 4\), \(y(0) = 7\), and \(y(1) = 3\).

    22. \(y' = –2y\) for \(y(0) = 4\), \(y(0) = 7\), and \(y(1) = 3\).

    Exercise \(\PageIndex{23}\)

    The rate of growth of a population \(P(t)\) which starts with 3,000 people and increases by 4% per year is \(P '(t) = 0.0392 \cdot P(t)\). Solve the differential equation and use the solution to estimate the population in 20 years.

    Exercise \(\PageIndex{24}\)

    The rate of growth of a population \(P(t)\) which starts with 5,000 people and increases by 3% per year is \(P '(t) = 0.0296 \cdot P(t)\). Solve the differential equation and use the solution to estimate the population in 20 years.

    Exercise \(\PageIndex{25}\)

    A manufacturer estimates that she can sell a maximum of 130 thousand cell phones in a city. By advertising heavily, her total sales grow at a rate proportional to the distance below this upper limit. If she enters a new market, and after 6 months her total sales are 59 thousand phones, find a formula for the total sales (in thousands) \(t\) months after entering the market, and use this to estimate the total sales at the end of the first year.

    Exercise \(\PageIndex{26}\)

    The temperature of a turkey in the oven will grow like limited growth. The turkey starts out at 40 degrees Fahrenheit, and is placed into a 350 degree oven. After 30 minutes, the turkey's temperature has risen to 55 degrees. How long will it take until the turkey's temperature reaches 165 degrees?

    Exercise \(\PageIndex{27}\)

    A new cell phone is introduced into the market. It is predicted that sales will grow logistically. The manufacturer estimates that they can sell a maximum of 100 thousand cell phones. After 44 thousand cell phones have been sold, sales are increasing by 4 thousand phones per month. Use this to estimate the total sales at the end of the first year.

    Exercise \(\PageIndex{28}\)

    Biologists stocked a lake with 400 fish and estimated the carrying capacity of the lake to be 8000 fish. The number of fish tripled in the first year. How long will it take the population to increase to 4000?


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