Worksheet: Method of Undetermined Coefficients

In this worksheet, we will practice solving a linear nonhomogeneous differential equation with constant coefficients by using the method of undetermined coefficients.

Q1:

Solve ddοŠ¨οŠ¨οŠ©ο—π‘¦π‘₯+9𝑦=9𝑒.

  • A𝑦=𝐢3π‘₯+𝐢3π‘₯+12π‘’οŠ§οŠ¨οŠ©ο—cossin
  • B𝑦=𝐢𝑒+𝐢𝑒+12π‘’οŠ§οŠ©ο—οŠ¨οŠ±οŠ©ο—οŠ©ο—
  • C𝑦=𝐢3π‘₯+𝐢3π‘₯βˆ’12π‘’οŠ§οŠ¨οŠ©ο—cossin
  • D𝑦=𝐢3π‘₯+𝐢3π‘₯+9π‘’οŠ§οŠ¨οŠ©ο—cossin
  • E𝑦=𝐢3π‘₯+𝐢3π‘₯+π‘’οŠ§οŠ¨οŠ©ο—cossin

Q2:

Solve ddddοŠ¨οŠ¨οŠ¨π‘¦π‘₯βˆ’10𝑦π‘₯+25𝑦=25π‘₯+5π‘₯+17.

  • A𝑦=𝐢𝑒+𝐢π‘₯𝑒+25π‘₯+5π‘₯+17οŠ§οŠ«ο—οŠ¨οŠ«ο—οŠ¨
  • B𝑦=𝐢𝑒+𝐢𝑒+25π‘₯+5π‘₯+17οŠ§οŠ«ο—οŠ¨οŠ«ο—οŠ¨
  • C𝑦=𝐢𝑒+𝐢𝑒+π‘₯+π‘₯+1οŠ§οŠ«ο—οŠ¨οŠ«ο—οŠ¨
  • D𝑦=𝐢𝑒+𝐢π‘₯π‘’βˆ’π‘₯+π‘₯+1οŠ§οŠ«ο—οŠ¨οŠ«ο—οŠ¨
  • E𝑦=𝐢𝑒+𝐢π‘₯𝑒+π‘₯+π‘₯+1οŠ§οŠ«ο—οŠ¨οŠ«ο—οŠ¨

Q3:

Solve ddddcosοŠ¨οŠ¨π‘¦π‘₯βˆ’2𝑦π‘₯βˆ’3𝑦=82π‘₯.

  • A𝑦=𝐢𝑒+πΆπ‘’βˆ’32652π‘₯+56652π‘₯οŠ§οŠ©ο—οŠ¨οŠ±ο—cossin
  • B𝑦=𝐢𝑒+πΆπ‘’βˆ’32652π‘₯+56652π‘₯οŠ§οŠ±οŠ©ο—οŠ¨ο—sincos
  • C𝑦=𝐢𝑒+𝐢𝑒+82π‘₯οŠ§οŠ©ο—οŠ¨οŠ±ο—cos
  • D𝑦=𝐢𝑒+𝐢𝑒+32652π‘₯βˆ’56652π‘₯οŠ§οŠ©ο—οŠ¨οŠ±ο—sincos
  • E𝑦=𝐢𝑒+𝐢𝑒+32652π‘₯+56652π‘₯οŠ§οŠ©ο—οŠ¨οŠ±ο—sincos

Q4:

Solve ddcosοŠ¨οŠ¨π‘¦π‘₯βˆ’9𝑦=63π‘₯.

  • A𝑦=𝐢𝑒+πΆπ‘’βˆ’133π‘₯οŠ§οŠ©ο—οŠ¨οŠ±οŠ©ο—cos
  • B𝑦=𝐢𝑒+𝐢𝑒+133π‘₯οŠ§οŠ©ο—οŠ¨οŠ±οŠ©ο—cos
  • C𝑦=𝐢𝑒+𝐢𝑒+63π‘₯οŠ§οŠ©ο—οŠ¨οŠ±οŠ©ο—cos
  • D𝑦=𝐢𝑒+πΆπ‘’βˆ’33π‘₯οŠ§οŠ©ο—οŠ¨οŠ±οŠ©ο—cos
  • E𝑦=𝐢𝑒+πΆπ‘’βˆ’133π‘₯οŠ§οŠ©ο—οŠ¨οŠ±οŠ©ο—sin

Q5:

Solve ddοŠ¨οŠ¨οŠ¨ο—οŠ¨π‘¦π‘₯+4𝑦=4𝑒+4π‘₯.

  • A𝑦=𝐢2π‘₯+𝐢2π‘₯+12𝑒+π‘₯βˆ’12οŠ§οŠ¨οŠ¨ο—οŠ¨cossin
  • B𝑦=𝐢2π‘₯+𝐢2π‘₯+𝑒+π‘₯βˆ’1οŠ§οŠ¨οŠ¨ο—οŠ¨cossin
  • C𝑦=𝐢2π‘₯+𝐢2π‘₯+2𝑒+π‘₯βˆ’12οŠ§οŠ¨οŠ¨ο—οŠ¨cossin
  • D𝑦=𝐢𝑒+𝐢𝑒+12𝑒+π‘₯βˆ’12οŠ§οŠ¨ο—οŠ¨οŠ±οŠ¨ο—οŠ¨ο—οŠ¨
  • E𝑦=𝐢2π‘₯+𝐢2π‘₯+12𝑒+π‘₯+12οŠ§οŠ¨οŠ¨ο—οŠ¨cossin

Q6:

Solve the third-order differential equation π‘¦β€²β€²β€²βˆ’2𝑦′′+𝑦′=2𝑒+2π‘₯ under the conditions 𝑦(0)=0, 𝑦′(0)=0, and 𝑦′′′(0)=0.

  • A𝑦=π‘₯+4π‘₯+4+ο€Ήπ‘₯βˆ’4ο…π‘’οŠ¨οŠ¨ο—
  • B𝑦=π‘₯+4π‘₯+4+ο€Ήπ‘₯+4ο…π‘’οŠ¨οŠ¨ο—
  • C𝑦=π‘₯+4π‘₯+4+ο€Ήπ‘₯βˆ’4ο…π‘’οŠ¨οŠ¨οŠ±ο—
  • D𝑦=π‘₯+4π‘₯+4+ο€Ήπ‘₯+4ο…π‘’οŠ¨οŠ¨οŠ±ο—

Q7:

Find the general solution for the ordinary differential equation π‘¦β€²β€²βˆ’2𝑦′+𝑦=2(π‘₯)cos with constant coefficients whose right-hand side is nonzero.

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

Q8:

Solve ddddοŠ¨οŠ¨π‘¦π‘₯βˆ’2𝑦π‘₯βˆ’3𝑦=2π‘₯+1.

  • A𝑦=𝐢𝑒+πΆπ‘’βˆ’23π‘₯+19οŠ§οŠ©ο—οŠ¨οŠ±ο—
  • B𝑦=𝐢𝑒+𝐢𝑒+23π‘₯+19οŠ§οŠ©ο—οŠ¨οŠ±ο—
  • C𝑦=𝐢𝑒+πΆπ‘’βˆ’23π‘₯βˆ’19οŠ§οŠ©ο—οŠ¨οŠ±ο—
  • D𝑦=𝐢𝑒+𝐢𝑒+2π‘₯+1οŠ§οŠ±οŠ©ο—οŠ¨ο—
  • E𝑦=𝐢𝑒+πΆπ‘’βˆ’32π‘₯βˆ’23οŠ§οŠ©ο—οŠ¨οŠ±ο—

Q9:

Solve ddddοŠ¨οŠ¨οŠ¨οŠ±ο—π‘¦π‘₯+2𝑦π‘₯+2𝑦=π‘₯𝑒.

  • A𝑦=𝑒(𝐢π‘₯+𝐢π‘₯)+π‘₯π‘’βˆ’2π‘’οŠ±ο—οŠ§οŠ¨οŠ¨οŠ±ο—οŠ±ο—cossin
  • B𝑦=𝑒(𝐢π‘₯+𝐢π‘₯)+π‘₯𝑒+2π‘’οŠ±ο—οŠ§οŠ¨οŠ¨οŠ±ο—οŠ±ο—cossin
  • C𝑦=𝑒(𝐢π‘₯+𝐢π‘₯)+π‘₯π‘’βˆ’12π‘’οŠ±ο—οŠ§οŠ¨οŠ¨οŠ±ο—οŠ±ο—cossin
  • D𝑦=𝑒(𝐢π‘₯+𝐢π‘₯)+π‘₯π‘’βˆ’2π‘’ο—οŠ§οŠ¨οŠ¨οŠ±ο—οŠ±ο—cossin
  • E𝑦=𝑒(𝐢π‘₯+𝐢π‘₯)+2π‘₯π‘’βˆ’π‘’οŠ±ο—οŠ§οŠ¨οŠ¨οŠ±ο—οŠ±ο—cossin

Q10:

Solve ddddοŠ¨οŠ¨οŠ¨ο—π‘¦π‘₯βˆ’2𝑦π‘₯βˆ’3𝑦=𝑒.

  • A𝑦=𝐢𝑒+𝐢𝑒+13π‘’οŠ§οŠ©ο—οŠ¨οŠ±ο—οŠ¨ο—
  • B𝑦=𝐢𝑒+πΆπ‘’βˆ’13π‘’οŠ§οŠ±οŠ©ο—οŠ¨ο—οŠ¨ο—
  • C𝑦=𝐢𝑒+πΆπ‘’βˆ’3π‘’οŠ§οŠ©ο—οŠ¨οŠ±ο—οŠ¨ο—
  • D𝑦=𝐢𝑒+πΆπ‘’βˆ’13π‘’οŠ§οŠ©ο—οŠ¨οŠ±ο—οŠ¨ο—
  • E𝑦=𝐢𝑒+𝐢𝑒+π‘’οŠ§οŠ©ο—οŠ¨οŠ±ο—οŠ¨ο—

Q11:

Solve the following differential equation under the conditions 𝑦(0)=βˆ’1 and 𝑦′(0)=βˆ’2: 𝑦′′+3𝑦′+2𝑦=10(π‘₯)sin.

  • A𝑦=𝑒+𝑒+(π‘₯)βˆ’3(π‘₯)οŠ±ο—οŠ¨ο—sincos
  • B𝑦=𝑒+𝑒+(π‘₯)βˆ’3(π‘₯)ο—οŠ¨ο—sincos
  • C𝑦=𝑒+𝑒+(π‘₯)βˆ’3(π‘₯)ο—οŠ±οŠ¨ο—sincos
  • D𝑦=𝑒+𝑒+(π‘₯)βˆ’3(π‘₯)οŠ±ο—οŠ±οŠ¨ο—sincos

Q12:

Solve ddοŠ¨οŠ¨οŠ©οŠ¨π‘¦π‘₯+4𝑦=4π‘₯+8π‘₯+18π‘₯+20.

  • A𝑦=𝐢2π‘₯+𝐢2π‘₯+4π‘₯+8π‘₯+18π‘₯+20cossin
  • B𝑦=𝐢2π‘₯+𝐢2π‘₯+4π‘₯+3π‘₯+2π‘₯+1cossin
  • C𝑦=𝐢2π‘₯+𝐢2π‘₯+4π‘₯+8π‘₯βˆ’6π‘₯+4cossin
  • D𝑦=𝐢2π‘₯+𝐢2π‘₯+π‘₯+2π‘₯+3π‘₯+4cossin
  • E𝑦=𝐢𝑒+𝐢𝑒+π‘₯+2π‘₯+3π‘₯+4οŠ§οŠ¨ο—οŠ¨οŠ±οŠ¨ο—οŠ©οŠ¨

Q13:

Solve ddddοŠ¨οŠ¨οŠ±ο—π‘¦π‘₯βˆ’3𝑦π‘₯βˆ’4𝑦=𝑒.

  • A𝑦=𝐢𝑒+𝐢𝑒+15π‘₯π‘’οŠ§οŠͺο—οŠ¨οŠ±ο—οŠ±ο—
  • B𝑦=𝐢𝑒+πΆπ‘’βˆ’5π‘₯π‘’οŠ§οŠͺο—οŠ¨οŠ±ο—οŠ±ο—
  • C𝑦=𝐢𝑒+πΆπ‘’βˆ’15π‘₯π‘’οŠ§οŠͺο—οŠ¨οŠ±ο—οŠ±ο—
  • D𝑦=𝐢𝑒+πΆπ‘’βˆ’15π‘₯π‘’οŠ§οŠ±οŠͺο—οŠ¨ο—οŠ±ο—
  • E𝑦=𝐢𝑒+πΆπ‘’βˆ’15π‘’οŠ§οŠͺο—οŠ¨οŠ±ο—οŠ±ο—

Q14:

Solve ddsinοŠ¨οŠ¨π‘¦π‘₯+4𝑦=82π‘₯.

  • A𝑦=𝐢𝑒+πΆπ‘’βˆ’2π‘₯2π‘₯οŠ§οŠ¨ο—οŠ¨οŠ±οŠ¨ο—cos
  • B𝑦=𝐢2π‘₯+𝐢2π‘₯+2π‘₯2π‘₯cossincos
  • C𝑦=𝐢2π‘₯+𝐢2π‘₯βˆ’2π‘₯2π‘₯cossincos
  • D𝑦=𝐢2π‘₯+𝐢2π‘₯+2π‘₯2π‘₯cossinsin
  • E𝑦=𝐢𝑒+𝐢𝑒+2π‘₯2π‘₯οŠ§οŠ¨ο—οŠ¨οŠ±οŠ¨ο—cos

Q15:

Solve ddddοŠ¨οŠ¨ο—π‘¦π‘₯βˆ’2𝑦π‘₯+𝑦=6𝑒.

  • A𝑦=𝐢𝑒+𝐢π‘₯𝑒+3π‘₯π‘’οŠ§ο—οŠ¨ο—ο—
  • B𝑦=𝐢𝑒+𝐢π‘₯𝑒+3π‘’οŠ§ο—οŠ¨ο—ο—
  • C𝑦=𝐢𝑒+𝐢π‘₯𝑒+6π‘₯π‘’οŠ§ο—οŠ¨ο—ο—
  • D𝑦=𝐢𝑒+𝐢π‘₯𝑒+3π‘₯π‘’οŠ§ο—οŠ¨ο—οŠ¨ο—
  • E𝑦=𝐢𝑒+𝐢𝑒+3π‘’οŠ§ο—οŠ¨ο—ο—

Q16:

Find the general solution for the following differential equation using the method of undetermined coefficients: 𝑦+3𝑦+2𝑦=𝑒,𝑖=βˆ’1.οŽ˜οŽ˜οŽ˜οƒο—οŠ¨

  • A𝑦=𝑐𝑒+𝑐𝑒+110ο€Ήπ‘’βˆ’3π‘–π‘’ο…οŠ§οŠ±οŠ¨ο—οŠ¨οŠ±ο—οƒο—οŠ±οƒο—
  • B𝑦=𝑐𝑒+𝑐𝑒+110𝑒+3π‘–π‘’ο…οŠ§οŠ±οŠ¨ο—οŠ¨οŠ±ο—οƒο—οƒο—
  • C𝑦=𝑐𝑒+𝑐𝑒+110ο€Ήπ‘’βˆ’3π‘–π‘’ο…οŠ§οŠ±οŠ¨ο—οŠ¨οŠ±ο—οƒο—οƒο—
  • D𝑦=𝑐𝑒+𝑐𝑒+110ο€Ήπ‘’βˆ’3π‘–π‘’ο…οŠ§οŠ¨ο—οŠ¨οŠ±ο—οƒο—οƒο—

Q17:

Solve ddcosοŠ¨οŠ¨π‘¦π‘₯+𝑦=π‘₯π‘₯.

  • A𝑦=𝐢π‘₯+𝐢π‘₯βˆ’14ο€Ήπ‘₯π‘₯+π‘₯π‘₯ο…οŠ§οŠ¨οŠ¨cossincossin
  • B𝑦=𝐢π‘₯+𝐢π‘₯+4ο€Ήπ‘₯π‘₯+π‘₯π‘₯ο…οŠ§οŠ¨οŠ¨cossincossin
  • C𝑦=𝐢π‘₯+𝐢π‘₯+π‘₯π‘₯+π‘₯π‘₯cossincossin
  • D𝑦=𝐢π‘₯+𝐢π‘₯+14ο€Ήπ‘₯π‘₯+π‘₯π‘₯ο…οŠ§οŠ¨οŠ¨cossincossin
  • E𝑦=𝐢π‘₯+𝐢π‘₯+14ο€Ήπ‘₯π‘₯+π‘₯π‘₯ο…οŠ§οŠ¨οŠ¨cossincossin

Q18:

Solve ddsinοŠ¨οŠ¨π‘¦π‘₯+4𝑦=16βˆ’53π‘₯.

  • A𝑦=𝐢2π‘₯+𝐢2π‘₯+43π‘₯+1cossinsin
  • B𝑦=𝐢𝑒+πΆπ‘’βˆ’3π‘₯+4οŠ§οŠ¨ο—οŠ¨οŠ±οŠ¨ο—sin
  • C𝑦=𝐢2π‘₯+𝐢2π‘₯+3π‘₯+4cossinsin
  • D𝑦=𝐢2π‘₯+𝐢2π‘₯βˆ’5133π‘₯+4cossinsin
  • E𝑦=𝐢2π‘₯+𝐢2π‘₯βˆ’3π‘₯+4cossinsin

Q19:

Which of the following is a particular solution to the differential equation 𝑓(π‘₯)+𝑓′(π‘₯)+𝑓′′(π‘₯)=9π‘₯+8π‘₯βˆ’4π‘₯+2.

  • A𝑓(π‘₯)=9π‘₯βˆ’19π‘₯βˆ’20π‘₯+60
  • B𝑓(π‘₯)=9π‘₯+35π‘₯βˆ’20π‘₯βˆ’48
  • C𝑓(π‘₯)=27π‘₯+16π‘₯βˆ’4
  • D𝑓(π‘₯)=9π‘₯4+8π‘₯3βˆ’2π‘₯+2π‘₯οŠͺ

Q20:

Solve ddddsinοŠ¨οŠ¨π‘¦π‘₯βˆ’2𝑦π‘₯+5𝑦=10π‘₯.

  • A𝑦=𝑒(𝐢2π‘₯+𝐢2π‘₯)+2π‘₯+π‘₯ο—οŠ§οŠ¨cossinsincos
  • B𝑦=𝑒(𝐢2π‘₯+𝐢2π‘₯)+10π‘₯ο—οŠ§οŠ¨cossinsin
  • C𝑦=𝑒(𝐢2π‘₯+𝐢2π‘₯)βˆ’2π‘₯+π‘₯ο—οŠ§οŠ¨cossinsincos
  • D𝑦=𝑒(𝐢2π‘₯+𝐢2π‘₯)+2π‘₯βˆ’π‘₯ο—οŠ§οŠ¨cossinsincos
  • E𝑦=𝑒(𝐢2π‘₯+𝐢2π‘₯)+π‘₯+2π‘₯ο—οŠ§οŠ¨cossinsincos

Q21:

Find the general solution for the following nonhomogeneous ordinary differential equation: 𝑦′′+2𝑦′+𝑦=π‘₯π‘’οŠ¨οŠ±ο—.

  • A𝑦=𝑐𝑒+𝑐π‘₯𝑒+π‘₯𝑒12οŠ§οŠ±ο—οŠ¨οŠ±ο—οŠͺ
  • B𝑦=𝑐𝑒+𝑐π‘₯𝑒+π‘₯𝑒12οŠ§οŠ±ο—οŠ¨ο—οŠͺοŠ±ο—
  • C𝑦=𝑐𝑒+𝑐π‘₯𝑒+π‘₯𝑒12οŠ§ο—οŠ¨οŠ±ο—οŠͺοŠ±ο—
  • D𝑦=𝑐𝑒+𝑐π‘₯𝑒+π‘₯𝑒12οŠ§οŠ±ο—οŠ¨οŠ±ο—οŠͺοŠ±ο—

Q22:

Find the particular solution π‘¦οŒ and the complementary function π‘¦οŒΌ for the following ordinary differential equation using the method of inverse operators: π‘¦β€²β€²βˆ’π‘¦=π‘₯+3π‘₯βˆ’4.

  • A𝑦=βˆ’ο€Ήπ‘₯+9π‘₯βˆ’4ο…οŒοŠ©, 𝑦=𝑐+π‘π‘’οŒΌοŠ§οŠ¨ο—
  • B𝑦=βˆ’ο€Ήπ‘₯+9π‘₯βˆ’4ο…οŒοŠ©, 𝑦=𝑐𝑒+π‘π‘’οŒΌοŠ§ο—οŠ¨οŠ±ο—
  • C𝑦=βˆ’ο€Ήπ‘₯+9π‘₯βˆ’4ο…οŒοŠ©, 𝑦=𝑐𝑒+π‘π‘’οŒΌοŠ§ο—οŠ¨οŠ±ο—
  • D𝑦=π‘₯+9π‘₯βˆ’4, 𝑦=𝑐𝑒+π‘οŒΌοŠ§ο—οŠ±ο—οŠ¨

Q23:

Solve ddddοŠ¨οŠ¨οŠ¨π‘¦π‘₯βˆ’2𝑦π‘₯+4𝑦=π‘₯.

  • A𝑦=π‘’ο€»πΆβˆš3π‘₯+𝐢√3π‘₯+14π‘₯βˆ’14π‘₯ο—οŠ§οŠ¨οŠ¨cossin
  • B𝑦=π‘’ο€»πΆβˆš3π‘₯+𝐢√3π‘₯+π‘₯ο—οŠ§οŠ¨οŠ¨cossin
  • C𝑦=π‘’ο€»πΆβˆš3π‘₯+𝐢√3π‘₯+14π‘₯+14π‘₯+14ο—οŠ§οŠ¨οŠ¨cossin
  • D𝑦=π‘’ο€»πΆβˆš3π‘₯+𝐢√3π‘₯+14π‘₯ο—οŠ§οŠ¨οŠ¨cossin
  • E𝑦=π‘’ο€»πΆβˆš3π‘₯+𝐢√3π‘₯+14π‘₯+14π‘₯ο—οŠ§οŠ¨οŠ¨cossin

Q24:

Solve ddsinοŠ¨οŠ¨ο—π‘¦π‘₯βˆ’4𝑦=π‘₯+3π‘₯𝑒.

  • A𝑦=𝐢𝑒+𝐢𝑒+π‘₯+3π‘₯π‘’οŠ§οŠ¨ο—οŠ¨οŠ±οŠ¨ο—ο—sin
  • B𝑦=𝐢𝑒+πΆπ‘’βˆ’15π‘₯+π‘₯𝑒+23π‘’οŠ§οŠ¨ο—οŠ¨οŠ±οŠ¨ο—ο—ο—sin
  • C𝑦=𝐢𝑒+πΆπ‘’βˆ’15π‘₯βˆ’π‘₯π‘’βˆ’23π‘’οŠ§οŠ¨ο—οŠ¨οŠ±οŠ¨ο—ο—ο—cos
  • D𝑦=𝐢𝑒+πΆπ‘’βˆ’13π‘₯βˆ’π‘₯π‘’βˆ’23π‘’οŠ§οŠ¨ο—οŠ¨οŠ±οŠ¨ο—ο—ο—sin
  • E𝑦=𝐢𝑒+πΆπ‘’βˆ’15π‘₯βˆ’π‘₯π‘’βˆ’23π‘’οŠ§οŠ¨ο—οŠ¨οŠ±οŠ¨ο—ο—ο—sin

Q25:

Solve ddcosοŠ¨οŠ¨π‘¦π‘₯+9𝑦=52π‘₯.

  • A𝑦=𝐢3π‘₯+𝐢3π‘₯+2π‘₯cossinsin
  • B𝑦=𝐢3π‘₯+𝐢3π‘₯βˆ’2π‘₯cossincos
  • C𝑦=𝐢𝑒+𝐢𝑒+52π‘₯οŠ§οŠ©ο—οŠ¨οŠ±οŠ©ο—cos
  • D𝑦=𝐢3π‘₯+𝐢3π‘₯+5132π‘₯cossincos
  • E𝑦=𝐢3π‘₯+𝐢3π‘₯+2π‘₯cossincos

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