Worksheet: Solving First-Order Differential Equations by Substitution

In this worksheet, we will practice using a suitable substitution to solve some first-order ordinary differential equations.

Q1:

Solve the differential equation d d 𝑦 𝑥 = ( 𝑥 + 𝑦 + 3 ) .

  • A 𝑦 = 𝐶 + 𝑥 3 t a n
  • B 𝑦 = ( 𝑥 + 𝐶 ) 𝑥 3 t a n h
  • C 𝑦 = ( 𝑥 + 𝐶 ) 𝑥 + 3 t a n
  • D 𝑦 = ( 𝑥 + 𝐶 ) 𝑥 3 t a n
  • E 𝑦 = ( 𝑥 + 𝐶 ) + 𝑥 + 3 t a n h

Q2:

Solve the differential equation 𝑦 = ( 4 𝑥 + 𝑦 ) .

  • A 𝑦 = 2 ( 4 𝑥 + 𝐶 ) + 4 𝑥 t a n
  • B 𝑦 = 2 ( 2 𝑥 + 𝐶 ) 4 𝑥 c o t
  • C 𝑦 = 2 ( 4 𝑥 + 𝐶 ) 4 𝑥 c o t
  • D 𝑦 = 2 ( 2 𝑥 + 𝐶 ) 4 𝑥 t a n
  • E 𝑦 = 2 ( 4 𝑥 + 𝐶 ) 4 𝑥 t a n

Q3:

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

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

Q4:

Solve the differential equation ( 𝑥 + 𝑦 ) 𝑦 = 1 .

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

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