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 d d 𝑦 𝑥 + 𝑦 𝑥 = 𝑘 𝑦 , where 𝑥 > 0 .

  • A 𝑦 = 1 𝑘 𝑥 C l n
  • B 𝑦 = 1 𝑥 ( 𝑘 𝑥 ) C l n
  • C 𝑦 = 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 + 𝑥 𝑦 = 𝑥 𝑦 , 𝑦 0 .

  • 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 d d c o t c s c 𝑦 𝑥 = 𝑦 𝑥 + 𝑘 𝑦 𝑥 .

  • A 𝑦 = 𝑥 2 𝑘 𝑥 + s i n c o s C and 𝑦 = 𝑥 2 𝑘 𝑥 + s i n c o s C
  • B 𝑦 = 𝑥 2 𝑘 𝑥 + s i n c o s C and 𝑦 = 𝑥 2 𝑘 𝑥 + s i n c o s C
  • C 𝑦 = 𝑥 2 𝑘 𝑥 s i n C c o s and 𝑦 = 𝑥 2 𝑘 𝑥 s i n C c o s
  • D 𝑦 = 2 𝑘 𝑥 + 𝑥 c o s C s i n and 𝑦 = 2 𝑘 𝑥 + 𝑥 c o s C s i n
  • E 𝑦 = 2 𝑘 𝑥 + 𝑥 c o s C s i n

Q4:

Find the nontrivial solution to the differential equation d d 𝑦 𝑥 + 𝑦 3 = 𝑘 𝑒 𝑦 , 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 d d 𝑦 𝑥 𝑘 𝑦 𝑥 = 𝑥 𝑦 where 𝑘 2 .

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

Q6:

Find the nontrivial solution to the differential equation d d c o s 𝑦 𝑥 + 2 𝑦 𝑥 = 𝑥 𝑦 𝑥 .

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

Q7:

Find any nontrivial solutions to the differential equation d d 𝑦 𝑥 + 𝑦 𝑥 = 𝑥 𝑦 .

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

Q8:

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

  • A 𝑦 = 𝑥 ( 𝐹 ( 𝑥 ) ) C
  • B 𝑦 = 𝑥 ( 𝐹 ( 𝑥 ) ) C
  • C 𝑦 = 1 𝑥 ( 𝐹 ( 𝑥 ) ) C
  • D 𝑦 = 1 𝑥 ( + 𝐹 ( 𝑥 ) ) C
  • E 𝑦 = 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 2 𝑦 𝑥 + 𝑦 𝑥 = 𝑦 𝑓 ( 𝑥 ) 𝑥 d d t a n c o s .

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

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