Lesson Worksheet: Bernoulli’s Differential Equation Mathematics

Start Practising

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.

  • A𝑦=1−𝑘𝑥Cln
  • B𝑦=𝑥(−𝑘𝑥)Cln
  • C𝑦=1𝑥(−𝑘𝑥)Cln
  • D𝑦=1𝑥(−𝑘𝑥)Cln
  • E𝑦=1𝑥(+𝑘𝑥)Cln

Q2:

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

  • A𝑦=𝑥2𝑘𝑥+sincosC and 𝑦=−𝑥2𝑘𝑥+sincosC
  • B𝑦=2𝑘𝑥+𝑥cosCsin and 𝑦=−2𝑘𝑥+𝑥cosCsin
  • C𝑦=𝑥2𝑘𝑥+sincosC and 𝑦=−𝑥2𝑘𝑥+sincosC
  • D𝑦=2𝑘𝑥+𝑥cosCsin
  • E𝑦=𝑥−2𝑘𝑥sinCcos and 𝑦=−𝑥−2𝑘𝑥sinCcos

Q3:

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

  • A𝑦=𝑒(−3𝑘𝑥)C
  • B𝑦=𝑒(+3𝑘𝑥)C
  • C𝑦=𝑒(−3𝑘𝑥)C
  • D𝑦=𝑒(+3𝑘𝑥)C
  • E𝑦=𝑒(−3𝑘𝑥)C

Q4:

Find the non-trivial solution to the differential equation dd𝑦𝑥−𝑘𝑦𝑥=𝑥𝑦 where 𝑘≠−2.

  • A𝑦=(𝑘+1)𝑥+𝑥C
  • B𝑦=(𝑘+2)𝑥−𝑥C
  • C𝑦=𝑥−𝑥C
  • D𝑦=(𝑘+2)𝑥+𝑥C
  • E𝑦=(𝑘+2)𝑥−𝑥C

Q5:

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

  • A𝑦=1𝑥(𝑥+)sinC
  • B𝑦=1𝑥(𝑥+)cosC
  • C𝑦=1𝑥(𝑥+)sinC
  • D𝑦=1𝑥(2𝑥+)sinC
  • E𝑦=𝑥(𝑥+)sinC

Q6:

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

  • A𝑦=2𝑥+𝑥C
  • B𝑦=2𝑥−𝑥C and 𝑦=𝑥−2𝑥C
  • C𝑦=12𝑥+𝑥C and 𝑦=−12𝑥+𝑥C
  • D𝑦=1√2𝑥+𝑥C and 𝑦=−1√2𝑥+𝑥C
  • E𝑦=1√𝑥−2𝑥C and 𝑦=−1√𝑥−2𝑥C

Q7:

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.

  • A𝑦=1𝑥(−𝐹(𝑥))C
  • B𝑦=𝑥(−𝐹(𝑥))C
  • C𝑦=𝑥(−𝐹(𝑥))C
  • D𝑦=𝑥(+𝐹(𝑥))C
  • E𝑦=1𝑥(+𝐹(𝑥))C

Q8:

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.

  • A𝑦=𝑥−𝐹(𝑥)cosC and 𝑦=−𝑥−𝐹(𝑥)cosC
  • B𝑦=𝑥𝐹(𝑥)−cosC and 𝑦=𝑥−𝐹(𝑥)cosC
  • C𝑦=𝑥+𝐹(𝑥)cosC
  • D𝑦=𝑥+𝐹(𝑥)cosC and 𝑦=−𝑥+𝐹(𝑥)cosC
  • E𝑦=−𝐹(𝑥)𝑥Ccos

Practice Means Progress

Boost your grades with free daily practice questions. Download Nagwa Practice today!

scan me!

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.