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Section 1.3 – Classification of Differential Equations

Directions. The following are review problems for the section. It is recommended you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it. 
  1. Given the following differential equations, classify each as an ordinary differential equation, partial differential equation, give the order. If the equation is an ordinary differential equation, say whether the equation is linear or non linear.
    1. \(\dfrac{d y}{d x}=3y+x^2\)
    2. \(5\dfrac{d^4 y}{d x^4}+y=x(x-1)\)
    3. \(\dfrac{\partial N}{\partial t}=\dfrac{\partial ^2N}{\partial r^2}+\frac{1}{r}\dfrac{\partial N}{\partial r}+kN\)
    4. \(\dfrac{d x}{d t}=x^2-t\)
    5. \((1+y^2)y''+ty'+y=e^t\)​

      \[
      \begin{array}{|c|c|c|c|c|}
      \hline
      Part& ODE& PDE & Order & Linear/Non Linear \\
      \hline 
      (a)&\checkmark &  & 1 & Linear\\
      \hline
      (b)&\checkmark&&4&Linear \\
      \hline
      (c)&&\checkmark&2&Linear \\
      \hline
      (d)&\checkmark&&1&Non Linear \\
      \hline
      (e)&\checkmark&&2&Non Linear\\
      \hline
      \end{array}
      \]

      To see the full video page and find related videos, click the following link.
      MATH 308 WIR22A V4

  2.  
    1. Show that \(f(x)=(x^2+Ax+B)e^{-x}\) is a solution to \[y''+2y'+y=2e^{-x}\] for all real numbers \(A\) and \(B\).
    2. Find a solution that satisfies the initial condition \(y(0)=3\) and \(y'(0)=1\).​

      1. For verification that \(f(x)\) is a solution for part (a), see the video.
      2. \(y=(x^2+4x+3)e^{-x}\)

      To see the full video page and find related videos, click the following link.
      MATH 308 WIR22A V5

  3. Determine for which values of \(r\) the function \(t^r\) is a solution of the differential equation
    \[t^2y''-4ty'+4y=0, \quad t>0\]

    \(r=1,4\)

    To see the full video page and find related videos, click the following link.
    MATH 308 WIR22A V6

  4. For which values of \(r\) is the function \((x-1)e^{-rx}\) a solution to \(y''-6y'+9y=0\)?

    \(r=-3\)


    To see the full video page and find related videos, click the following link.
    MATH 308 WIR22A V7