Worksheet: Bernoulli’s Differential Equation

In this worksheet, we will practice solving Bernoulli’s differential equation, which has the form y’ + p(x) y = q(x) yⁿ, by reducing it to a linear differential equation.

Q1:

Find the nontrivial solution to the differential equation dd𝑦π‘₯+𝑦π‘₯=π‘˜π‘¦οŠ¨, where π‘₯>0.

  • A𝑦=1βˆ’π‘˜π‘₯Cln
  • B𝑦=π‘₯(βˆ’π‘˜π‘₯)Cln
  • C𝑦=1π‘₯(βˆ’π‘˜π‘₯)Cln
  • D𝑦=1π‘₯(βˆ’π‘˜π‘₯)Cln
  • E𝑦=1π‘₯(+π‘˜π‘₯)Cln

Q2:

Find any nontrivial solutions to the differential equation ddcotcsc𝑦π‘₯=𝑦π‘₯+π‘˜π‘¦π‘₯.

  • A𝑦=π‘₯2π‘˜π‘₯+sincosC and 𝑦=βˆ’π‘₯2π‘˜π‘₯+sincosC
  • B𝑦=ο„Ÿ2π‘˜π‘₯+π‘₯cosCsin and 𝑦=βˆ’ο„Ÿ2π‘˜π‘₯+π‘₯cosCsin
  • C𝑦=ο„Ÿπ‘₯2π‘˜π‘₯+sincosC and 𝑦=βˆ’ο„Ÿπ‘₯2π‘˜π‘₯+sincosC
  • D𝑦=2π‘˜π‘₯+π‘₯cosCsin
  • E𝑦=ο„Ÿπ‘₯βˆ’2π‘˜π‘₯sinCcos and 𝑦=βˆ’ο„Ÿπ‘₯βˆ’2π‘˜π‘₯sinCcos

Q3:

Find the nontrivial solution to the differential equation dd𝑦π‘₯+𝑦3=π‘˜π‘’π‘¦ο—οŠͺ, where π‘₯>0.

  • A𝑦=𝑒(βˆ’3π‘˜π‘₯)ο‘οŽ’οŽ οŽ’C
  • B𝑦=𝑒(+3π‘˜π‘₯)οŠ±οŠ©ο—οŠ±οŠ©C
  • C𝑦=𝑒(βˆ’3π‘˜π‘₯)οŠ±οŠ±ο‘οŽ’οŽ οŽ’C
  • D𝑦=𝑒(+3π‘˜π‘₯)οŠ±οŠ±ο‘οŽ’οŽ οŽ’C
  • E𝑦=𝑒(βˆ’3π‘˜π‘₯)C

Q4:

Find the non-trivial solution to the differential equation dd𝑦π‘₯βˆ’π‘˜π‘¦π‘₯=π‘₯π‘¦οŠ¨ where π‘˜β‰ βˆ’2.

  • A𝑦=(π‘˜+1)π‘₯+π‘₯ο‡ο‡οŠ°οŠ§C
  • B𝑦=(π‘˜+2)π‘₯βˆ’π‘₯ο‡ο‡οŠ°οŠ¨C
  • C𝑦=π‘₯βˆ’π‘₯ο‡ο‡οŠ°οŠ¨C
  • D𝑦=(π‘˜+2)π‘₯+π‘₯ο‡ο‡οŠ°οŠ¨C
  • E𝑦=(π‘˜+2)π‘₯βˆ’π‘₯ο‡ο‡οŠ°οŠ§C

Q5:

Find the nontrivial solution to the differential equation ddcos𝑦π‘₯+2𝑦π‘₯=βˆ’π‘₯𝑦π‘₯.

  • A𝑦=1π‘₯(π‘₯+)sinC
  • B𝑦=1π‘₯(π‘₯+)cosC
  • C𝑦=1π‘₯(π‘₯+)sinC
  • D𝑦=1π‘₯(2π‘₯+)sinC
  • E𝑦=π‘₯(π‘₯+)sinC

Q6:

Find any nontrivial solutions to the differential equation dd𝑦π‘₯+𝑦π‘₯=π‘₯π‘¦οŠ©.

  • A𝑦=2π‘₯+π‘₯C
  • B𝑦=2π‘₯βˆ’π‘₯C and 𝑦=π‘₯βˆ’2π‘₯C
  • C𝑦=12π‘₯+π‘₯C and 𝑦=βˆ’12π‘₯+π‘₯C
  • D𝑦=1√2π‘₯+π‘₯C and 𝑦=βˆ’1√2π‘₯+π‘₯C
  • E𝑦=1√π‘₯βˆ’2π‘₯C and 𝑦=βˆ’1√π‘₯βˆ’2π‘₯C

Q7:

Given that there exists a function 𝐹 such that 𝐹′(π‘₯)=𝑓(π‘₯), find the nontrivial solution to the differential equation π‘₯𝑦π‘₯+𝑦=𝑦π‘₯𝑓(π‘₯)dd.

  • A𝑦=1π‘₯(βˆ’πΉ(π‘₯))C
  • B𝑦=π‘₯(βˆ’πΉ(π‘₯))C
  • C𝑦=π‘₯(βˆ’πΉ(π‘₯))C
  • D𝑦=π‘₯(+𝐹(π‘₯))C
  • E𝑦=1π‘₯(+𝐹(π‘₯))C

Q8:

Given that there exists a function 𝐹 such that 𝐹′(π‘₯)=𝑓(π‘₯), find any nontrivial solutions to the differential equation 2𝑦π‘₯+𝑦π‘₯=𝑦𝑓(π‘₯)π‘₯ddtancos.

  • A𝑦=ο„žπ‘₯βˆ’πΉ(π‘₯)cosC and 𝑦=βˆ’ο„žπ‘₯βˆ’πΉ(π‘₯)cosC
  • B𝑦=π‘₯𝐹(π‘₯)βˆ’cosC and 𝑦=π‘₯βˆ’πΉ(π‘₯)cosC
  • C𝑦=π‘₯+𝐹(π‘₯)cosC
  • D𝑦=ο„žπ‘₯+𝐹(π‘₯)cosC and 𝑦=βˆ’ο„žπ‘₯+𝐹(π‘₯)cosC
  • E𝑦=βˆ’πΉ(π‘₯)π‘₯Ccos

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