Worksheet: 2nd-Order Linear Homogeneous Differential Equations with Constant Coefficients

In this worksheet, we will practice solving first-order differential equations.

Q1:

Consider the differential equation 𝑦 + 𝑦 = 0 . Suppose that a student determined the solution to be 𝑦 = 𝑒 𝑥 . Based upon this information, is the student correct?

  • ANo, the student should have determined that there is no solution.
  • BNo, the student should have determined the solution to be 𝑦 = 𝑒 𝑥 .
  • CYes, the student did not make any errors.
  • DNo, the student should have determined the solution to be 𝑦 = 𝑒 𝑥 .

Q2:

Find the general solution for the following higher-order differential equation: 𝑦 + 9 𝑦 + 2 7 𝑦 + 2 7 𝑦 = 0 .

  • A 𝑦 = 𝑐 𝑒 + 𝑐 𝑥 𝑒 + 𝑐 𝑥 𝑒 1 𝑥 2 2 𝑥 3 2 3 𝑥
  • B 𝑦 = 𝑐 𝑥 𝑒 + 𝑐 𝑥 𝑒 + 𝑐 𝑥 𝑒 1 3 𝑥 2 2 3 𝑥 3 3 3 𝑥
  • C 𝑦 = 𝑐 + 𝑐 𝑥 𝑒 + 𝑐 𝑥 𝑒 1 2 𝑥 3 2 2 𝑥
  • D 𝑦 = 𝑐 𝑒 + 𝑐 𝑥 𝑒 + 𝑐 𝑥 𝑒 1 3 𝑥 2 3 𝑥 3 2 3 𝑥

Q3:

The one-dimensional time-independent Schrodinger equation is given as d d 𝜓 𝑥 = 2 𝑚 [ 𝑈 ( 𝑥 ) 𝐸 ] 𝜓 ,

where 𝜓 is a wave function which describes the displacement 𝑥 of a single particle of mass 𝑚 , 𝐸 is the total energy, 𝑈 is the potential energy, and is a known constant. Since 𝑈 ( 𝑥 ) = 0 for the particle-in-a-box model, where 0 𝑥 𝑎 , this second-order differential equation becomes 𝜓 = 𝛼 𝜓 , 𝛼 = 2 𝑚 𝐸 . w h e r e

Find the general solution for this differential equation.

  • A 𝑦 = ( 𝑐 + 𝑐 ) 𝑒
  • B 𝑦 = ( 𝑐 + 𝑐 ) 𝑒
  • C 𝑦 = 𝑐 𝑒 + 𝑐 𝑒
  • D 𝑦 = 𝑐 𝑒 + 𝑐 𝑒

Q4:

Find the general solution for the following homogeneous ordinary differential equation with constant coefficients: 𝑦 2 𝑦 + 𝑦 = 0 .

  • A 𝑦 = ( 𝑐 𝑥 + 𝑐 ) 𝑒
  • B 𝑦 = 𝑐 𝑥 𝑒 + 𝑐 𝑒
  • C 𝑦 = 𝑐 𝑒 + 𝑐 𝑥 𝑒
  • D 𝑦 = ( 𝑐 𝑥 + 𝑐 ) 𝑒

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