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Worksheet: General First-Order Differential Equations and Solutions

Q1:

Consider the differential equation 𝑦 β€² + 𝑦 = 0 . Suppose that a student determined the solution to be 𝑦 = 𝑒 π‘₯ . 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 = βˆ’ 𝑒 π‘₯ .
  • CYes, the student did not make any errors.
  • DNo, the student should have determined the solution to be 𝑦 = 𝑒 βˆ’ π‘₯ .

Q2:

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

  • A 𝑦 = 𝑐 𝑒 + 𝑐 π‘₯ 𝑒 + 𝑐 π‘₯ 𝑒 1 βˆ’ π‘₯ 2 βˆ’ 2 π‘₯ 3 2 βˆ’ 3 π‘₯
  • B 𝑦 = 𝑐 π‘₯ 𝑒 + 𝑐 π‘₯ 𝑒 + 𝑐 π‘₯ 𝑒 1 βˆ’ 3 π‘₯ 2 2 βˆ’ 3 π‘₯ 3 3 βˆ’ 3 π‘₯
  • C 𝑦 = 𝑐 + 𝑐 π‘₯ 𝑒 + 𝑐 π‘₯ 𝑒 1 2 βˆ’ π‘₯ 3 2 βˆ’ 2 π‘₯
  • D 𝑦 = 𝑐 𝑒 + 𝑐 π‘₯ 𝑒 + 𝑐 π‘₯ 𝑒 1 βˆ’ 3 π‘₯ 2 βˆ’ 3 π‘₯ 3 2 βˆ’ 3 π‘₯

Q3:

The one-dimensional time-independent Schrodinger equation is given as

where πœ“ is a wave function which describes the displacement π‘₯ of a single particle of mass π‘š , 𝐸 is the total energy, π‘ˆ is the potential energy, and ℏ is a known constant. Since π‘ˆ ( π‘₯ ) = 0 for the particle-in-a-box model, where 0 ≀ π‘₯ ≀ π‘Ž , this second-order differential equation becomes

Find the general solution for this differential equation.

  • A 𝑦 = ( 𝑐 + 𝑐 ) 𝑒 1 2 𝛼 π‘₯
  • B 𝑦 = ( 𝑐 + 𝑐 ) 𝑒 1 2 βˆ’ 𝛼 π‘₯
  • C 𝑦 = 𝑐 𝑒 + 𝑐 𝑒 1 2 𝛼 π‘₯ βˆ’ 𝛼 π‘₯
  • D 𝑦 = 𝑐 𝑒 + 𝑐 𝑒 1 𝛼 π‘₯ 2 βˆ’ 𝛼 π‘₯

Q4:

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

  • A 𝑦 = ( 𝑐 π‘₯ + 𝑐 ) 𝑒 1 2 βˆ’ π‘₯
  • B 𝑦 = 𝑐 π‘₯ 𝑒 + 𝑐 𝑒 1 π‘₯ 2 βˆ’ π‘₯
  • C 𝑦 = 𝑐 𝑒 + 𝑐 π‘₯ 𝑒 1 π‘₯ 2 βˆ’ π‘₯
  • D 𝑦 = ( 𝑐 π‘₯ + 𝑐 ) 𝑒 1 2 π‘₯