Worksheet: First-Order Linear Differential Equations

In this worksheet, we will practice solving linear first-order differential equations.

Q1:

Solve the differential equation d d 𝑦 π‘₯ + 𝑦 = π‘₯ .

  • A 𝑦 = π‘₯ βˆ’ 1 + 𝑒 C  
  • B 𝑦 = π‘₯ 𝑒 2 + 𝑒      C
  • C 𝑦 = π‘₯ 𝑒 2 + 𝑒     C
  • D 𝑦 = π‘₯ βˆ’ 1 + 𝑒 C 
  • E 𝑦 = π‘₯ + 1 + 𝑒 C  

Q2:

Solve the differential equation π‘₯ 𝑦 π‘₯ + 𝑦 = π‘₯ π‘₯ d d l n , where π‘₯ > 0 , subject to the condition 𝑦 ( 1 ) = 0 .

  • A 𝑦 = π‘₯ 2 π‘₯ + π‘₯ 4 + 1 4 π‘₯ l n
  • B 𝑦 = π‘₯ 2 π‘₯ βˆ’ π‘₯ 4 + 1 2 π‘₯ l n
  • C 𝑦 = π‘₯ π‘₯ βˆ’ π‘₯ 4 + 1 4 π‘₯ l n
  • D 𝑦 = π‘₯ 2 π‘₯ βˆ’ π‘₯ 4 + 1 4 π‘₯ l n
  • E 𝑦 = π‘₯ 4 π‘₯ βˆ’ π‘₯ 4 + 1 4 π‘₯ l n

Q3:

Solve the differential equation π‘₯ 𝑦 π‘₯ + π‘₯ 𝑦 = 1  d d , where π‘₯ > 0 , subject to the condition 𝑦 ( 1 ) = 2 .

  • A 𝑦 = π‘₯ + 2 π‘₯ l n
  • B 𝑦 = π‘₯ + 2 2 π‘₯ l n
  • C 𝑦 = π‘₯ βˆ’ 2 + 2 π‘₯ l n l n
  • D 𝑦 = π‘₯ π‘₯ l n
  • E 𝑦 = βˆ’ 1 π‘₯ + 3 

Q4:

Solve the differential equation π‘₯ 𝑦 π‘₯ = 𝑦 + π‘₯ π‘₯ d d s i n  subject to the condition 𝑦 ( πœ‹ ) = 0 .

  • A 𝑦 = βˆ’ π‘₯ π‘₯ βˆ’ π‘₯ c o s
  • B 𝑦 = βˆ’ π‘₯ π‘₯ + π‘₯ c o s
  • C 𝑦 = π‘₯ π‘₯ c o s
  • D 𝑦 = π‘₯ π‘₯ βˆ’ π‘₯ c o s
  • E 𝑦 = π‘₯ π‘₯ + π‘₯ c o s

Q5:

Solve the differential equation 𝑑 𝑒 𝑑 = 𝑑 + 3 𝑒 d d  subject to the condition 𝑒 ( 2 ) = 4 .

  • A 𝑒 = βˆ’ 𝑑 + 𝑑 
  • B 𝑒 = βˆ’ 𝑑 + 𝑑  
  • C 𝑒 = βˆ’ 𝑑 βˆ’ 𝑑  
  • D 𝑒 = 𝑑 
  • E 𝑒 = 𝑑 5 + 1 2 8 5 𝑑  

Q6:

Solve the differential equation 𝑑 𝑦 𝑑 + 3 𝑑 𝑦 = √ 1 + 𝑑   d d , where 𝑑 > 0 .

  • A 𝑦 = 1 3 ο€Ή 1 + 𝑑  𝑑 + 𝑑        C
  • B 𝑦 = 1 3 ο€Ή 1 + 𝑑  𝑑 + 𝑑        C
  • C 𝑦 = ο€Ή 1 + 𝑑  𝑑 + 𝑑        C
  • D 𝑦 = 1 3 ( 1 + 𝑑 ) 𝑑 + 𝑑       C
  • E 𝑦 = 1 3 ( 1 + 𝑑 ) 𝑑 + 𝑑       C

Q7:

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

  • A 𝑦 = 2 √ π‘₯ 3 + C
  • B 𝑦 = 2 5 √ π‘₯ + π‘₯  C
  • C 𝑦 = √ π‘₯ 2 + π‘₯ C
  • D 𝑦 = 2 5 √ π‘₯ + π‘₯  C
  • E 𝑦 = 2 √ π‘₯ 3 + π‘₯ C

Q8:

Solve the differential equation 2 π‘₯ 𝑦 π‘₯ + 𝑦 = 2 √ π‘₯ d d .

  • A 𝑦 = π‘₯ + π‘₯ C
  • B 𝑦 = √ π‘₯ + √ π‘₯  C
  • C 𝑦 = π‘₯ + √ π‘₯ C
  • D 𝑦 = √ π‘₯ + √ π‘₯ C
  • E 𝑦 = 1 + √ π‘₯ C

Q9:

Solve the differential equation ο€Ή π‘₯ + 1  𝑦 π‘₯ + 3 π‘₯ ( 𝑦 βˆ’ 1 ) = 0  d d subject to the condition 𝑦 ( 0 ) = 2 .

  • A 𝑦 = 1 + 1 ( π‘₯ + 1 )   
  • B 𝑦 = 1 + 1 ( π‘₯ + 1 )   
  • C 𝑦 = 1 βˆ’ 1 ( π‘₯ + 1 )   
  • D 𝑦 = 3 + 1 ( π‘₯ + 1 )   
  • E 𝑦 = 1 βˆ’ 5 √ 5 ( π‘₯ + 1 )   

Q10:

Is the differential equation d d c o s π‘Ÿ 𝑑 + 𝑑 π‘Ÿ = 𝑒   linear?

  • Ano
  • Byes

Q11:

Solve the differential equation π‘₯ 𝑦 π‘₯ βˆ’ 2 𝑦 = π‘₯ d d  , where π‘₯ > 0 .

  • A 1 4 π‘₯ + π‘₯   C
  • B 𝑦 = βˆ’ 1 2 π‘₯ ( π‘₯ + )  l n C
  • C 𝑦 = π‘₯ ( π‘₯ + )  C
  • D 𝑦 = βˆ’ π‘₯ ( π‘₯ + )  l n C
  • E 𝑦 = π‘₯ ( π‘₯ + )  l n C

Q12:

Solve the differential equation π‘₯ 𝑦 π‘₯ + 2 π‘₯ 𝑦 = π‘₯  d d l n subject to the condition 𝑦 ( 1 ) = 2 .

  • A 𝑦 = π‘₯ + 4 2 π‘₯ l n 
  • B 𝑦 = π‘₯ βˆ’ π‘₯ + 3 π‘₯ l n 
  • C 𝑦 = π‘₯ π‘₯ βˆ’ π‘₯ + 3 π‘₯ l n 
  • D 𝑦 = π‘₯ π‘₯ + 3 π‘₯ l n 
  • E 𝑦 = π‘₯ π‘₯ βˆ’ π‘₯ + 1 π‘₯ l n 

Q13:

Solve the differential equation 4 π‘₯ 𝑦 + π‘₯ 𝑦 π‘₯ = π‘₯  οŠͺ  d d s i n .

  • A 𝑦 = βˆ’ π‘₯ + 3 π‘₯ + 3 π‘₯ c o s c o s C  οŠͺ
  • B 𝑦 = π‘₯ βˆ’ 3 π‘₯ + 3 π‘₯ c o s c o s C  οŠͺ
  • C 𝑦 = βˆ’ π‘₯ + 3 π‘₯ + 3 π‘₯ c o s c o s C  οŠͺ
  • D 𝑦 = π‘₯ + 3 π‘₯ + 3 π‘₯ c o s c o s C  οŠͺ
  • E 𝑦 = π‘₯ βˆ’ π‘₯ + 3 π‘₯ c o s c o s C  οŠͺ

Q14:

Solve the differential equation 𝑑 𝑦 𝑑 + 3 𝑑 𝑦 = 𝑑   d d c o s subject to the condition 𝑦 ( πœ‹ ) = 0 .

  • A 𝑦 = 𝑑 + 1 𝑑 s i n 
  • B 𝑦 = 𝑑 𝑑 + 𝑑 + 1 𝑑 s i n c o s 
  • C 𝑦 = βˆ’ 𝑑 + 1 𝑑 s i n 
  • D 𝑦 = 𝑑 𝑑 s i n 
  • E 𝑦 = βˆ’ 𝑑 𝑑 s i n 

Q15:

Is the differential equation 𝑒 𝑒 = 𝑑 + √ 𝑑 𝑒 𝑑  d d linear?

  • Ayes
  • Bno

Q16:

Solve the differential equation 𝑑 𝑑 π‘Ÿ 𝑑 + π‘Ÿ = 𝑑 𝑒 l n d d  , where 𝑑 > 0 .

  • A π‘Ÿ = 𝑒 + 𝑑   C l n
  • B π‘Ÿ = 𝑒 + 2 𝑑  C l n
  • C π‘Ÿ = 1 + 𝑑 C l n
  • D π‘Ÿ = 𝑒 + 𝑑  C l n
  • E π‘Ÿ = 𝑒 + 2 𝑑   C l n

Q17:

Solve the differential equation d d 𝑦 π‘₯ βˆ’ 𝑦 = 𝑒  .

  • A 𝑦 = ( π‘₯ + ) 𝑒 C 
  • B 𝑦 = ( π‘₯ + ) 𝑒 C  
  • C 𝑦 = βˆ’ ( π‘₯ + ) 𝑒 C 
  • D 𝑦 = ( 𝑒 + ) 𝑒   C
  • E 𝑦 = ( π‘₯ + ) 𝑒 C  

Q18:

Is the differential equation d d t a n 𝑦 π‘₯ βˆ’ π‘₯ = 𝑦 π‘₯ linear?

  • Ayes
  • Bno

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