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 dd𝑦𝑥+𝑦𝑥=𝑘𝑦, where 𝑥>0.

  • A 𝑦 = 1 𝑘 𝑥 C l n
  • B 𝑦 = 𝑥 ( 𝑘 𝑥 ) C l n
  • C 𝑦 = 1 𝑥 ( 𝑘 𝑥 ) C l n
  • D 𝑦 = 1 𝑥 ( 𝑘 𝑥 ) C l n
  • E 𝑦 = 1 𝑥 ( + 𝑘 𝑥 ) C l n

Q2:

Find the general solution for the following Bernoulli differential equation: 𝑦+𝑥𝑦=𝑥𝑦,𝑦0.

  • A 𝑦 = 1 + 𝑐 𝑒
  • B 𝑦 = 1 + 𝑐 𝑒
  • C 𝑦 = 1 + 𝑐 𝑒
  • D 𝑦 = 1 + 𝑐 𝑒

Q3:

Find any nontrivial solutions to the differential equation ddcotcsc𝑦𝑥=𝑦𝑥+𝑘𝑦𝑥.

  • A 𝑦 = 𝑥 2 𝑘 𝑥 + s i n c o s C and 𝑦=𝑥2𝑘𝑥+sincosC
  • B 𝑦 = 2 𝑘 𝑥 + 𝑥 c o s C s i n and 𝑦=2𝑘𝑥+𝑥cosCsin
  • C 𝑦 = 𝑥 2 𝑘 𝑥 + s i n c o s C and 𝑦=𝑥2𝑘𝑥+sincosC
  • D 𝑦 = 2 𝑘 𝑥 + 𝑥 c o s C s i n
  • E 𝑦 = 𝑥 2 𝑘 𝑥 s i n C c o s and 𝑦=𝑥2𝑘𝑥sinCcos

Q4:

Find the nontrivial solution to the differential equation dd𝑦𝑥+𝑦3=𝑘𝑒𝑦, where 𝑥>0.

  • A 𝑦 = 𝑒 ( 3 𝑘 𝑥 ) C
  • B 𝑦 = 𝑒 ( + 3 𝑘 𝑥 ) C
  • C 𝑦 = 𝑒 ( 3 𝑘 𝑥 ) C
  • D 𝑦 = 𝑒 ( + 3 𝑘 𝑥 ) C
  • E 𝑦 = 𝑒 ( 3 𝑘 𝑥 ) C

Q5:

Find the non-trivial solution to the differential equation dd𝑦𝑥𝑘𝑦𝑥=𝑥𝑦 where 𝑘2.

  • A 𝑦 = ( 𝑘 + 1 ) 𝑥 + 𝑥 C
  • B 𝑦 = ( 𝑘 + 2 ) 𝑥 𝑥 C
  • C 𝑦 = 𝑥 𝑥 C
  • D 𝑦 = ( 𝑘 + 2 ) 𝑥 + 𝑥 C
  • E 𝑦 = ( 𝑘 + 2 ) 𝑥 𝑥 C

Q6:

Find the nontrivial solution to the differential equation ddcos𝑦𝑥+2𝑦𝑥=𝑥𝑦𝑥.

  • A 𝑦 = 1 𝑥 ( 𝑥 + ) s i n C
  • B 𝑦 = 1 𝑥 ( 𝑥 + ) c o s C
  • C 𝑦 = 1 𝑥 ( 𝑥 + ) s i n C
  • D 𝑦 = 1 𝑥 ( 2 𝑥 + ) s i n C
  • E 𝑦 = 𝑥 ( 𝑥 + ) s i n C

Q7:

Find any nontrivial solutions to the differential equation dd𝑦𝑥+𝑦𝑥=𝑥𝑦.

  • A 𝑦 = 2 𝑥 + 𝑥 C
  • B 𝑦 = 2 𝑥 𝑥 C and 𝑦=𝑥2𝑥C
  • C 𝑦 = 1 2 𝑥 + 𝑥 C and 𝑦=12𝑥+𝑥C
  • D 𝑦 = 1 2 𝑥 + 𝑥 C and 𝑦=12𝑥+𝑥C
  • E 𝑦 = 1 𝑥 2 𝑥 C and 𝑦=1𝑥2𝑥C

Q8:

Given that there exists a function 𝐹 such that 𝐹(𝑥)=𝑓(𝑥), find the nontrivial solution to the differential equation 𝑥𝑦𝑥+𝑦=𝑦𝑥𝑓(𝑥)dd.

  • A 𝑦 = 1 𝑥 ( 𝐹 ( 𝑥 ) ) C
  • B 𝑦 = 𝑥 ( 𝐹 ( 𝑥 ) ) C
  • C 𝑦 = 𝑥 ( 𝐹 ( 𝑥 ) ) C
  • D 𝑦 = 𝑥 ( + 𝐹 ( 𝑥 ) ) C
  • E 𝑦 = 1 𝑥 ( + 𝐹 ( 𝑥 ) ) C

Q9:

Given that there exists a function 𝐹 such that 𝐹(𝑥)=𝑓(𝑥), find any nontrivial solutions to the differential equation 2𝑦𝑥+𝑦𝑥=𝑦𝑓(𝑥)𝑥ddtancos.

  • A 𝑦 = 𝑥 𝐹 ( 𝑥 ) c o s C and 𝑦=𝑥𝐹(𝑥)cosC
  • B 𝑦 = 𝑥 𝐹 ( 𝑥 ) c o s C and 𝑦=𝑥𝐹(𝑥)cosC
  • C 𝑦 = 𝑥 + 𝐹 ( 𝑥 ) c o s C
  • D 𝑦 = 𝑥 + 𝐹 ( 𝑥 ) c o s C and 𝑦=𝑥+𝐹(𝑥)cosC
  • E 𝑦 = 𝐹 ( 𝑥 ) 𝑥 C c o s

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