Homework

Unit 1

Assignment 1

  1. Determine whether the given function is a solution to the differential equation:
    1. \(y=\frac{x^3}{3}+\pi\)   ,   \(\frac{dy}{dx}=x^2\)
    2. \(y=\frac{1}{(2-x)^2}\)   ,   \(y'=y^2\)
    3. \(y=\sqrt{4-x^2}\)   ,   \(y\frac{dy}{dx}+x=0\)
    4. \(x=\cos{\left(\sqrt{\frac{k}{m}}\cdot t\right)}\)   ,   \(m\ddot{x}+kx=0\)
    5. \(y=e^{-x}+e^x\)   ,   \(y''=y\)
    6. \(y=ax^2+bx+c\)   ,   \(\frac{d^3y}{dx^3}=0\)
  2. Use implicit differentiation to show that \(x^2+y^2=4\) is a solution to \(\frac{dy}{dx}=-\frac{x}{y}\).
  3. For which values of \(m\) is \(\phi(x)=e^{mx}\) a solution to \(\frac{d^2y}{dx^2}+6\frac{dy}{dx}+5y=0\)
  4. Solve the following differential equations by integrating or separation of variables:
    1. \(y'=x^2\)  ,    \(y(-3)=2\)
    2. \(\frac{dy}{dx}=\frac{y-1}{x+3}\)  ,    \((-1,0)\)
    3. \(x^2dx=-ydy\)
    4. \(\sin(2\theta)d\theta+\cos(y)dy=0\)
    5. \(\frac{dy}{dx}=\frac{4xy}{x^2+1}\)
    6. \(y''=1\)  ,    \(y(0)=1\)  ,  \(y'(0)=2\)

Assignment 2

  1. Draw a slope field for \(\frac{dy}{dx}=y(2-y)\)  for \(0\le x\le 4\) and \(0\le y\le 4\).  Draw the slope segments every 0.25 units.  Then, draw a couple of curves through the field.
  2. Use a slope field application to create a slope field for \(\frac{dy}{dx}=x-y\).  Use Euler's method to estimate a solution curve with the initial value, \(y(0)=1\) and \(\delta x = 0.2\).
  3. Use a slope field application to create a slope field for \(\frac{dy}{dx}=p(p-1)(2-p)\).  Sketch curves for the initial values \(y=4,\; y=1.5,\) and \(y=0.5\), when \(x=0\).

Assignment 3

  1. \(y'-\frac{1}{x}y=x^3\)
  2. \(\frac{dy}{dx}-\frac{2y}{x}=x^5\) , \(x>0\)
  3. \(y'-2xy=1\)  ,    \(y(0)=1\)
  4. \(y'-3x=0\)
  5. \(y'-3x=15\)
  6. \(y'+y=0\)  ,   \(y(-1)=3\)
  7. \(y'+y=e^-x\)  ,  \( y(-1)=3\)
  8. \(y'+y\sin(x)=0\)  ,  \( y(0)=0\)
  9. \(y'+y\sin(x)=kx\)  ,   \(y(0)=0\)
  10. Rewrite \(p(x)=a+\int_0^r t^2 p(t) dt\) as a differential equation and solve for \(p(r)\)
  11. \(\cos(x)\frac{dy}{dx}+y\,\sin(x)=2x\,\cos^2(x)\)  ,    \(y\left(\frac{\pi}{4}\right)=\frac{-15\sqrt{2}\pi^2}{32}\)
  12. Bernoulli Equations: \(\frac{dy}{dx}+2y=xy^{-2}\) .  Substitute \(v=y^3\) to convert to a linear differential equation.
  13. A tank is initially filled with 1000 L of water with 6 kg of salt. A brine containing 0.3 kg of salt per liter runs into the tank at 5 L/min.  The mixture is kept uniformly mixed and is flowing out at 6 L/min.  How much salt is in the tank after 8 minutes?

Assignment 4

  1. Solve: \((x+y^2)dx+(y-x^2)dy=0\)
  2. Solve: \((3x^2y^2+x^2)dx+(2x^3y+y^2)dy=0\)
  3. Solve: \((2x+ye^{xy})dx+(xe^{xy})dy=0\)
  4. Solve: \((3x^2+2xy^2)dx+(2x^2y)dy=0\) where \(y(2)=-3\)
  5. If \(y^2\sin(x)dx+yf(x)dy=0\) is an exact equation, then what could \(f(x)\) be?

Assignment 5

  1. Consider the linear differential equations \(\frac{dv}{dt}+\frac{k}{m}v=g\) ,  where \(v\) is the velocity of a falling object, \(g\) is the acceleration due to gravity, \(m\) is the mass of the object, and \(k\) is the friction constant due to air resistance.
    Solve the initial value problem, where \(v(0)=v_o\) and find the terminal velocity, \(v_l\)
  2. With the solution to problem 1 above for \(v\), solve the displacement integral for \(x\):

    \[x-x_0=\int_0^t vdt\]

  3. A body at a temperature of \(100^\circ\) is placed in a room of constant temperature.  If after 10 minutes the body has cooled to \(90^\circ\) and after 20 minutes to \(85^\circ\), then find the temperature of the room.
  4. More coming soon . . .

UNIT 2

Assignment 1

  1. What is the period of a non-zero solution to \(\ddot{x}+4x=0\)?
  2. Check that \(x_1=\cos(\omega t)\) and \(x_2=\sin(\omega t)\) are solutions to:

    \[\ddot{x}+\omega^2x=0\]

  3. Check that \(A\cos(\omega t+\phi)\) is also a solution to:

    \[\ddot{x}+\omega^2x=0\]

  4.  For what values of \(r\) is \(x=e^{rt}\) a solution to:

    \[\ddot{x}+kx=0\]

  5. Find the general solution for \(y''+4y'+5y=0\).
  6. Find the general solution for \(y''+2y'-3y=0\).  Then find a specific solution for the initial conditions \(y(0)=1\) and \(y'(0)=-1\)
  7. Find the general solution for \(\ddot{x}+2\dot{x}+2x=0\).
  8. Find the general solution for \(y''+2y'+5y=0\).  Then find a specific solution for the initial conditions \(y(0)=1\) and \(y'(0)=-1\)
  9. Find the general solution for \(y''+ 6y'+ 9y = 0\).

6 Responses to “Homework”

  1. andrew says:

    Is part 6 of question 4 the initial values for parts 3, 4, and 5?

    Thanks!

    Andrew

    • gary says:

      No, The differential equation is y''=1.
      Since it is a second order equation, two conditions are needed to find a specific solution. Here the initial conditions are y(0)=1 and y'(0)=2. So, at the point (0,1) the solution curve has a slope of 2.

  2. andrew says:

    What are the p's in part three of assigment 2?

  3. andrew says:

    For parts 2 and 3 of assignment 2, are we just supposed to do it on the computer, actually sketch it out on paper, or something else?

    Andrew

    • Gary says:

      If you are going to turn it in for a grade, then you could print it out, or copy it into a file, or email it, or copy it by hand. Otherwise, just look at it.

      Some of the applications that I posted also sketch Euler's method for estimating solutions.

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