Worksheet: Linear Homogeneous Differential Equations with Constant Coefficients

In this worksheet, we will practice solving second- and higher-order linear homogeneous differential equations with constant coefficients.

Q1:

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

  • ANo, the student should have determined the solution to be 𝑦=π‘’οŠ±ο—.
  • BNo, the student should have determined the solution to be 𝑦=βˆ’π‘’ο—.
  • CNo, the student should have determined that there is no solution.
  • DYes, the student did not make any errors.

Q2:

Find the general solution for the following higher-order differential equation: 𝑦′′′+9𝑦′′+27𝑦′+27𝑦=0.

  • A𝑦=𝑐𝑒+𝑐π‘₯𝑒+𝑐π‘₯π‘’οŠ§οŠ±ο—οŠ¨οŠ±οŠ¨ο—οŠ©οŠ¨οŠ±οŠ©ο—
  • B𝑦=𝑐𝑒+𝑐π‘₯𝑒+𝑐π‘₯π‘’οŠ§οŠ±οŠ©ο—οŠ¨οŠ±οŠ©ο—οŠ©οŠ¨οŠ±οŠ©ο—
  • C𝑦=𝑐+𝑐π‘₯𝑒+𝑐π‘₯π‘’οŠ§οŠ¨οŠ±ο—οŠ©οŠ¨οŠ±οŠ¨ο—
  • D𝑦=𝑐π‘₯𝑒+𝑐π‘₯𝑒+𝑐π‘₯π‘’οŠ§οŠ±οŠ©ο—οŠ¨οŠ¨οŠ±οŠ©ο—οŠ©οŠ©οŠ±οŠ©ο—

Q3:

Find the general solution for the following homogeneous ordinary differential equation with constant coefficients: π‘¦βˆ’2𝑦+𝑦=0.

  • A𝑦=𝑐𝑒+𝑐π‘₯π‘’οŠ§ο—οŠ¨οŠ±ο—
  • B𝑦=(𝑐π‘₯+𝑐)π‘’οŠ§οŠ¨οŠ±ο—
  • C𝑦=𝑐π‘₯𝑒+π‘π‘’οŠ§ο—οŠ¨οŠ±ο—
  • D𝑦=(𝑐π‘₯+𝑐)π‘’οŠ§οŠ¨ο—

Q4:

Apply the superposition principle to find the solution for π‘¦β€²β€²βˆ’π‘¦=0.

  • A𝑦=𝑐𝑒+π‘π‘’οŠ§ο—οŠ¨οŠ±ο—
  • B𝑦=𝑐π‘₯+π‘βˆšπ‘₯
  • C𝑦=𝑐π‘₯+𝑐π‘₯sincos
  • D𝑦=𝑐(π‘₯)+𝑐π‘₯lncos

Q5:

Find a particular solution for 𝑦=𝑦′′ which passes through the origin and through the point ο€Ό(2),34ln.

  • A𝑦=(π‘₯)cos
  • B𝑦=(π‘₯)cosh
  • C𝑦=(π‘₯)sinh
  • D𝑦=(π‘₯)sin

Q6:

Suppose there are differentiable functions 𝐢 and 𝑆, defined for all real numbers and satisfying 𝐢=π‘†οŽ˜ and 𝑆=𝐢.

From the derivatives, what can you deduce about πΆβˆ’π‘†οŠ¨οŠ¨?

  • AπΆβˆ’π‘†οŠ¨οŠ¨ is a constant function.
  • BπΆβˆ’π‘†οŠ¨οŠ¨ is an exponential function.
  • CπΆβˆ’π‘†οŠ¨οŠ¨ is a hyperbolic function.
  • DπΆβˆ’π‘†οŠ¨οŠ¨ always equals 0.
  • EπΆβˆ’π‘†οŠ¨οŠ¨ is a trigonometric function.

It is known that every function that satisfies 𝑦′′=𝑦 is a combination π‘Žπ‘’+π‘π‘’ο—οŠ±ο— for some constants π‘Ž and 𝑏. Find all functions 𝐢 satisfying 𝐢(0)=5 and πΆβˆ’π‘†=24, where 𝑆=𝐢′.

  • A𝐢(π‘₯)=2π‘’βˆ’3π‘’ο—οŠ±ο— and 𝐢(π‘₯)=βˆ’3𝑒+2π‘’ο—οŠ±ο—
  • B𝐢(π‘₯)=βˆ’2π‘’βˆ’3π‘’ο—οŠ±ο— and 𝐢(π‘₯)=βˆ’3π‘’βˆ’2π‘’ο—οŠ±ο—
  • C𝐢(π‘₯)=βˆ’2𝑒+3π‘’ο—οŠ±ο— and 𝐢(π‘₯)=3π‘’βˆ’2π‘’ο—οŠ±ο—
  • D𝐢(π‘₯)=2𝑒+3π‘’ο—οŠ±ο— and 𝐢(π‘₯)=3𝑒+2π‘’ο—οŠ±ο—
  • E𝐢(π‘₯)=5𝑒+5π‘’ο—οŠ±ο— and 𝐢(π‘₯)=βˆ’5π‘’βˆ’5π‘’ο—οŠ±ο—

Sketching graphs of the functions above suggest that 𝐢(π‘₯)=𝐴(π‘₯βˆ’π‘˜)cosh for some constants 𝐴 and π‘˜. Find these constants, where π‘˜>0.

  • A𝐴=2√6, π‘˜=32ln
  • B𝐴=2√6, π‘˜=3βˆ’22lnln
  • C𝐴=5, π‘˜=32ln
  • D𝐴=5, π‘˜=22ln
  • E𝐴=5, π‘˜=3βˆ’22lnln

Q7:

Find the particular solution for the following homogeneous differential equation: π‘¦βˆ’81𝑦=0(οŠͺ) where 𝑦(0)=4, 𝑦′(0)=12, 𝑦′′(0)=βˆ’18 and 𝑦′′′(0)=βˆ’162.

  • A𝑦=𝑒+3(3π‘₯)+5(3π‘₯)οŠ±οŠ©ο—cossin
  • B𝑦=𝑒+5(3π‘₯)+3(3π‘₯)οŠ©ο—cossin
  • C𝑦=𝑒+5(3π‘₯)+3(3π‘₯)οŠ±οŠ©ο—cossin
  • D𝑦=𝑒+3(3π‘₯)+5(3π‘₯)οŠ©ο—cossin

Q8:

Find the solution for the following second-order ordinary differential equation: 𝑦′′+4𝑦′+5𝑦=0, where 𝑦(0)=1 and 𝑦′(0)=βˆ’1.

  • A𝑦=𝑒(π‘₯)+𝑒(π‘₯)οŠ±οŠ¨ο—οŠ¨ο—cossin
  • B𝑦=𝑒(2π‘₯)+𝑒(2π‘₯)cossin
  • C𝑦=𝑒(π‘₯)+𝑒(π‘₯)οŠ¨ο—οŠ¨ο—cossin
  • D𝑦=𝑒(π‘₯)+𝑒(π‘₯)οŠ±οŠ¨ο—οŠ±οŠ¨ο—cossin

Q9:

Solve the following differential equation under the conditions 𝑦(0)=1 and 𝑦(0)=0: 2𝑦π‘₯+2𝑦π‘₯+𝑦=0.dddd

  • A𝑦=𝑒π‘₯2+𝑒π‘₯2ο‡οŠ±οŠ±ο‘οŽ‘ο‘οŽ‘cossin
  • B𝑦=𝑒π‘₯2+𝑒π‘₯2ο‡οŠ±ο‘οŽ‘ο‘οŽ‘cossin
  • C𝑦=𝑒π‘₯2+𝑒π‘₯2ο‡ο‘οŽ‘ο‘οŽ‘cossin
  • D𝑦=𝑒π‘₯2+𝑒π‘₯2ο‡ο‘οŽ‘ο‘οŽ‘cossin

Q10:

Solve the following differential equation under the conditions 𝑦(0)=1 and 𝑦′(0)=1: ddοŠ¨οŠ¨π‘¦π‘₯+4𝑦=0.

  • A𝑦=(2π‘₯)+12(2π‘₯)sincos
  • B𝑦=(π‘₯)+12(π‘₯)cossin
  • C𝑦=(2π‘₯)+12(2π‘₯)cossin
  • D𝑦=(π‘₯)+12(π‘₯)sincos

Q11:

Solve the following differential equation under the conditions 𝑦(0)=4 and 𝑦′(0)=4: ddddοŠ¨οŠ¨π‘¦π‘₯+6𝑦π‘₯+9𝑦=0.

  • A𝑦=16π‘₯𝑒+4π‘’οŠ©ο—οŠ±οŠ©ο—
  • B𝑦=16π‘₯4𝑒+π‘’οŠ±οŠ©ο—οŠ©ο—
  • C𝑦=16π‘₯𝑒+π‘’οŠ©ο—οŠ©ο—
  • D𝑦=16π‘₯4𝑒+4π‘’οŠ±οŠ©ο—οŠ±οŠ©ο—

Q12:

Solve the fourth-order differential equation π‘¦βˆ’π‘¦=0(οŠͺ) under the conditions 𝑦(0)=3, 𝑦′(0)=1, 𝑦′′(0)=βˆ’1, and 𝑦′′′(0)=βˆ’3.

  • A𝑦=𝑒+2(π‘₯)+2(π‘₯)cossin
  • B𝑦=π‘’βˆ’2(π‘₯)+2(π‘₯)cossin
  • C𝑦=𝑒+2(π‘₯)+2(π‘₯)οŠ±ο—cossin
  • D𝑦=𝑒+2(π‘₯)βˆ’2(π‘₯)οŠ±ο—cossin

Q13:

The functions 𝑦=π‘’οŠ§ο—, 𝑦=π‘’οŠ¨οŠ±ο—, and 𝑦=π‘’οŠ©οŠ±οŠ¨ο— are three linearly independent solutions of the differential equation 𝑦+2π‘¦β€²β€²βˆ’π‘¦β€²βˆ’2𝑦=0(). Find a particular solution satisfying the initial conditions 𝑦(0)=1, 𝑦′(0)=2, and 𝑦′′(0)=0.

  • A𝑦=43π‘’βˆ’13π‘’ο—οŠ±ο—
  • B𝑦=43𝑒+13π‘’ο—οŠ±οŠ¨ο—
  • C𝑦=43π‘’βˆ’13π‘’ο—οŠ±οŠ¨ο—
  • D𝑦=𝑒+π‘’βˆ’π‘’ο—οŠ±ο—οŠ±οŠ¨ο—
  • E𝑦=43𝑒+13π‘’ο—οŠ±ο—

Q14:

The functions 𝑦=π‘’οŠ§ο—, 𝑦=π‘’οŠ¨οŠ¨ο—, and 𝑦=π‘’οŠ©οŠ©ο— are three linearly independent solutions of the differential equation π‘¦βˆ’6𝑦′′+11π‘¦β€²βˆ’6𝑦=0(). Find a particular solution satisfying the initial conditions 𝑦(0)=0, 𝑦′(0)=0, and 𝑦′′(0)=3.

  • A𝑦=32π‘’βˆ’3𝑒+32π‘’ο—οŠ¨ο—οŠ©ο—
  • B𝑦=βˆ’34𝑒+32π‘’βˆ’34π‘’ο—οŠ¨ο—οŠ©ο—
  • C𝑦=12π‘’βˆ’π‘’+12π‘’ο—οŠ¨ο—οŠ©ο—
  • D𝑦=βˆ’3𝑒+3π‘’ο—οŠ¨ο—
  • E𝑦=βˆ’π‘’+π‘’ο—οŠ¨ο—

Q15:

The functions 𝑦=π‘’οŠ§ο—, 𝑦=π‘₯π‘’οŠ¨ο—, and 𝑦=π‘₯π‘’οŠ©οŠ¨ο— are three linearly independent solutions of the differential equation π‘¦βˆ’3𝑦′′+3π‘¦β€²βˆ’π‘¦=0(). Find a particular solution satisfying the initial conditions 𝑦(0)=2, 𝑦′(0)=0, and 𝑦′′(0)=0.

  • A𝑦=2𝑒+2π‘₯𝑒+π‘₯π‘’ο—ο—οŠ¨ο—
  • B𝑦=2π‘’βˆ’2π‘₯𝑒+π‘₯π‘’ο—ο—οŠ¨ο—
  • C𝑦=2π‘’βˆ’2π‘₯π‘’βˆ’π‘₯π‘’ο—ο—οŠ¨ο—
  • D𝑦=βˆ’2𝑒+2π‘₯π‘’βˆ’π‘₯π‘’ο—ο—οŠ¨ο—
  • E𝑦=2𝑒+2π‘₯π‘’βˆ’π‘₯π‘’ο—ο—οŠ¨ο—

Q16:

The functions 𝑦=π‘’οŠ§ο—, 𝑦=π‘’οŠ¨οŠ¨ο—, and 𝑦=π‘₯π‘’οŠ©οŠ¨ο— are three linearly independent solutions of the differential equation π‘¦βˆ’5𝑦+8π‘¦βˆ’4𝑦=0(). Find a particular solution satisfying the initial conditions 𝑦(0)=1, 𝑦(0)=4, and 𝑦(0)=0.

  • A𝑦=12𝑒+13π‘’βˆ’10π‘₯π‘’ο—οŠ¨ο—οŠ¨ο—
  • B𝑦=12π‘’βˆ’13𝑒+10π‘₯π‘’ο—οŠ¨ο—οŠ¨ο—
  • C𝑦=βˆ’12𝑒+13π‘’βˆ’10π‘₯π‘’ο—οŠ¨ο—οŠ¨ο—
  • D𝑦=βˆ’12π‘’βˆ’13π‘’βˆ’10π‘₯π‘’ο—οŠ¨ο—οŠ¨ο—
  • E𝑦=βˆ’12𝑒+13𝑒+10π‘₯π‘’ο—οŠ¨ο—οŠ¨ο—

Q17:

The functions 𝑦=1, 𝑦=3π‘₯cos, and 𝑦=3π‘₯sin are three linearly independent solutions of the differential equation 𝑦+9𝑦′=0(). Find a particular solution satisfying the initial conditions 𝑦(0)=3, 𝑦′(0)=βˆ’1, and 𝑦′′(0)=2.

  • A𝑦=299βˆ’293π‘₯βˆ’133π‘₯cossin
  • B𝑦=299+293π‘₯βˆ’133π‘₯cossin
  • C𝑦=299+293π‘₯+133π‘₯cossin
  • D𝑦=299βˆ’133π‘₯βˆ’293π‘₯cossin
  • E𝑦=299+133π‘₯βˆ’293π‘₯cossin

Q18:

The functions 𝑦=π‘’οŠ§ο—, 𝑦=𝑒π‘₯οŠ¨ο—cos, and 𝑦=𝑒π‘₯οŠ©ο—sin are three linearly independent solutions of the differential equation π‘¦βˆ’3𝑦+4π‘¦βˆ’2𝑦=0(). Find a particular solution satisfying the initial conditions 𝑦(0)=1, 𝑦(0)=0, and 𝑦(0)=0.

  • A𝑦=βˆ’2π‘’βˆ’π‘’π‘₯βˆ’π‘’π‘₯cossin
  • B𝑦=2𝑒+𝑒π‘₯βˆ’π‘’π‘₯cossin
  • C𝑦=𝑒π‘₯+𝑒π‘₯cossin
  • D𝑦=2π‘’βˆ’π‘’π‘₯βˆ’π‘’π‘₯cossin
  • E𝑦=2𝑒+𝑒π‘₯+𝑒π‘₯cossin

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