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.


Solve the differential equation dd𝑦π‘₯=(π‘₯+𝑦+3).

  • A𝑦=(π‘₯+𝐢)βˆ’π‘₯βˆ’3tanh
  • B𝑦=(π‘₯+𝐢)+π‘₯+3tanh
  • C𝑦=(π‘₯+𝐢)βˆ’π‘₯+3tan
  • D𝑦=𝐢+π‘₯βˆ’3tan
  • E𝑦=(π‘₯+𝐢)βˆ’π‘₯βˆ’3tan


Solve the differential equation 𝑦′=(4π‘₯+𝑦).

  • A𝑦=2(2π‘₯+𝐢)βˆ’4π‘₯tan
  • B𝑦=2(4π‘₯+𝐢)+4π‘₯tan
  • C𝑦=2(2π‘₯+𝐢)βˆ’4π‘₯cot
  • D𝑦=2(4π‘₯+𝐢)βˆ’4π‘₯cot
  • E𝑦=2(4π‘₯+𝐢)βˆ’4π‘₯tan


Solve the differential equation 𝑦′=√π‘₯+𝑦+1.

  • A2√π‘₯+𝑦+1βˆ’2ο€Ί1+√π‘₯+𝑦+1=π‘₯+𝐢ln
  • Bln(π‘₯+𝑦+2)=π‘₯+𝐢
  • C2√π‘₯+𝑦+2=π‘₯+𝐢
  • D√π‘₯+𝑦+1+ο€Ί1+√π‘₯+𝑦+1=π‘₯+𝐢ln
  • E12√π‘₯+𝑦+2=π‘₯+𝐢


Solve the differential equation (π‘₯+𝑦)𝑦′=1.

  • A𝑦=(1+π‘₯+𝑦)+𝐢ln
  • B𝑦=βˆ’(1+π‘₯+𝑦)+𝐢ln
  • C𝑦=(1+π‘₯+𝑦)βˆ’2π‘₯+𝐢ln
  • D𝑦=(1βˆ’π‘₯βˆ’π‘¦)+𝐢ln
  • E𝑦=(1+π‘₯+𝑦)+2π‘₯+𝐢ln

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