Worksheet: Separable Differential Equations

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In this worksheet, we will practice identifying and solving separable differential equations.

Q1:

Solve the differential equation d d 𝑦 𝑥 + 𝑦 = 1 .

  • A 𝑦 = 𝑥 + 𝑒 C − 𝑥
  • B 𝑦 = 𝑥 𝑒 + 𝑒 − 𝑥 − 𝑥 C
  • C 𝑦 = 1 + 𝑒 C 𝑥
  • D 𝑦 = 1 + 𝑒 C − 𝑥
  • E 𝑦 = 𝑥 + 𝑒 C 𝑥

Q2:

Solve the differential equation d d 𝑦 𝑥 = − 5 𝑥 𝑦 .

  • A 𝑦 = 1 5 𝑥 + C or 𝑦 = 0
  • B 𝑦 = − 3 5 𝑥 + C or 𝑦 = 0
  • C 𝑦 = 1 1 5 𝑥 + C or 𝑦 = 0
  • D 𝑦 = 3 5 𝑥 + C or 𝑦 = 0
  • E 𝑦 = − 1 1 5 𝑥 + C or 𝑦 = 0

Q3:

Solve the differential equation d d l n 𝐻 𝑅 = 𝑅 𝐻 √ 1 + 𝑅 𝐻 .

  • A − 𝐻 𝐻 − 1 𝐻 = 1 2 1 + 𝑅 + l n C
  • B − 𝐻 𝐻 − 1 𝐻 = 2 3 1 + 𝑅 + l n C
  • C l n C 𝐻 𝐻 + 1 𝐻 = 1 3 1 + 𝑅 +
  • D − 𝐻 𝐻 − 1 𝐻 = 1 3 1 + 𝑅 + l n C
  • E − 𝐻 𝐻 + 1 𝐻 = 1 3 1 + 𝑅 + l n C

Q4:

Solve the differential equation d d s e c 𝜃 𝑡 = 𝑡 𝜃 𝜃 𝑒 .

  • A 𝜃 𝜃 = − 𝑒 2 + s i n C
  • B 𝜃 𝜃 + 𝜃 = 𝑒 + s i n c o s C
  • C − 𝜃 𝜃 − 𝜃 = − 𝑒 2 + s i n c o s C
  • D 𝜃 𝜃 + 𝜃 = − 𝑒 2 + s i n c o s C
  • E − 𝜃 𝜃 − 𝜃 = 𝑒 + s i n c o s C

Q5:

Find a relation between 𝑦 and 𝑥 , given that 𝑥 𝑦 𝑦 ′ = 𝑥 − 5 .

  • A 𝑦 = 2 𝑥 − 1 0 | 𝑥 | + l n C
  • B 𝑦 = 𝑥 − 5 | 𝑥 | + l n C
  • C 𝑦 = 𝑥 2 − 5 | 𝑥 | + l n C
  • D 𝑦 = 𝑥 − 1 0 | 𝑥 | + l n C
  • E 𝑦 = 2 𝑥 − 1 0 𝑥 + l n C

Q6:

Solve the following differential equation: d d 𝑝 𝑡 = 𝑡 𝑝 − 5 𝑝 + 𝑡 − 5 .

  • A 𝑝 = 𝑒 + 1 K
  • B 𝑝 = 𝑒 − 1 K
  • C 𝑝 = 𝑒 + 1 K
  • D 𝑝 = 𝑒 − 1 K
  • E 𝑝 = 𝑒 − 1 K

Q7:

Solve the following differential equation: ( 𝑒 − 5 ) 𝑦 ′ = 2 + 𝑥 c o s .

  • A 𝑒 − 5 𝑦 = 𝑥 + 𝑥 + s i n C
  • B 𝑒 − 5 𝑦 = 2 𝑥 − 𝑥 + s i n C
  • C 𝑒 − 5 = 2 𝑥 + 𝑥 + s i n C
  • D 𝑒 − 5 𝑦 = 2 𝑥 + 𝑥 + s i n C
  • E 𝑒 − 5 = 2 𝑥 − 𝑥 + s i n C

Q8:

Solve the differential equation 𝑦 + 𝑥 𝑒 = 0 ′ 𝑦 .

  • A 𝑦 = − 𝑥 + l n C 2
  • B 𝑦 = 𝑥 2 + l n C 2
  • C 𝑦 = − 2 𝑥 + l n C 2
  • D 𝑦 = − 𝑥 2 + l n C 2
  • E 𝑦 = 2 𝑥 + l n C 2

Q9:

Solve the differential equation d d 𝑦 𝑥 = − 5 𝑥 √ 𝑦 .

  • A √ 𝑦 = − 5 𝑥 + 2 C or 𝑦 = 0
  • B √ 𝑦 = − 5 𝑥 2 + 2 C or 𝑦 = 0
  • C 𝑦 = − 5 𝑥 4 + 2 2 C or 𝑦 = 0
  • D √ 𝑦 = − 5 𝑥 4 + 2 C or 𝑦 = 0
  • E 𝑦 = − 5 𝑥 2 + 2 2 C or 𝑦 = 0

Q10:

Solve the following differential equation by separating it:

  • A 𝑦 = ( | 𝑥 | + ) c o s l n C
  • B 𝑦 = ( | 𝑥 | + ) l n c o s C
  • C 𝑦 = ( | 𝑥 | + ) l n s i n C
  • D 𝑦 = ( | 𝑥 | + ) s i n l n C

Q11:

Find a 1-parameter family of solutions for the differential equation 𝑦 𝑦 = ( 𝑦 + 1 ) , 𝑦 ≠ − 1 .

  • A 1 𝑦 + 1 − | 𝑦 + 1 | = 𝑥 + 𝑐 l n
  • B − 1 𝑦 + 1 + | 𝑦 + 1 | = 𝑥 + 𝑐 l n
  • CThere is no solution.
  • D 1 𝑦 + 1 + | 𝑦 + 1 | = 𝑥 + 𝑐 l n

Q12:

Find the implicit solution to the following differential equation:

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

Q13:

Which of the following is a solution of 𝑥 + 𝑦 𝑦 ′ = 0 defined for all − 4 < 𝑥 < 4 ?

  • A 𝑦 = √ 1 6 + 𝑥 2
  • B 𝑦 = √ 4 − 𝑥 2
  • C 𝑦 = √ 4 + 𝑥 2
  • D 𝑦 = √ 1 6 − 𝑥 2

Q14:

Find a relation between 𝑢 and 𝑡 given that d d 𝑢 𝑡 = 1 + 𝑡 𝑢 𝑡 + 𝑢 𝑡 4 2 4 2 .

  • A 𝑢 5 + 𝑢 2 = − 1 𝑡 + 𝑡 + 5 2 3 C
  • B 𝑢 5 + 𝑢 2 = 1 𝑡 + 𝑡 3 + 5 2 3 C
  • C 𝑢 + 𝑢 = − 1 𝑡 + 𝑡 3 + 5 2 3 C
  • D 𝑢 5 + 𝑢 2 = − 1 𝑡 + 𝑡 3 + 5 2 3 C
  • E 𝑢 + 𝑢 = 1 𝑡 + 𝑡 3 + 5 2 3 C

Q15:

Solve the differential equation d d 𝑧 𝑡 + 𝑒 = 0 2 𝑡 + 2 𝑧 .

  • A 𝑧 = − 1 2 𝑒 2 + l n C 2 𝑡
  • B 𝑧 = − 1 2 2 𝑒 + l n C 2 𝑡
  • C 𝑧 = 1 2 𝑒 + l n C 2 𝑡
  • D 𝑧 = − 1 2 𝑒 + l n C 2 𝑡

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