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 dd𝑦π‘₯+𝑦=1.

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

Q2:

Solve the differential equation dd𝑦π‘₯=βˆ’5π‘₯π‘¦οŠ¨οŠ¨.

  • A 𝑦 = 1 1 5 π‘₯ +  C or 𝑦=0
  • B 𝑦 = βˆ’ 1 1 5 π‘₯ +  C or 𝑦=0
  • C 𝑦 = βˆ’ 3 5 π‘₯ +  C or 𝑦=0
  • D 𝑦 = 1 5 π‘₯ +  C or 𝑦=0
  • E 𝑦 = 3 5 π‘₯ +  C or 𝑦=0

Q3:

Solve the differential equation ddln𝐻𝑅=π‘…π»βˆš1+π‘…π»οŠ¨οŠ¨.

  • A βˆ’ 𝐻 𝐻 βˆ’ 1 𝐻 = 2 3 ο€Ή 1 + 𝑅  + l n C   
  • B βˆ’ 𝐻 𝐻 βˆ’ 1 𝐻 = 1 2 ο€Ή 1 + 𝑅  + l n C   
  • C l n C 𝐻 𝐻 + 1 𝐻 = 1 3 ο€Ή 1 + 𝑅  +   
  • D βˆ’ 𝐻 𝐻 βˆ’ 1 𝐻 = 1 3 ο€Ή 1 + 𝑅  + l n C   
  • E βˆ’ 𝐻 𝐻 + 1 𝐻 = 1 3 ο€Ή 1 + 𝑅  + l n C   

Q4:

Solve the differential equation ddsecπœƒπ‘‘=π‘‘πœƒπœƒπ‘’οοŽ‘.

  • A πœƒ πœƒ + πœƒ = 𝑒 + s i n c o s C   
  • B πœƒ πœƒ + πœƒ = βˆ’ 𝑒 2 + s i n c o s C   
  • C βˆ’ πœƒ πœƒ βˆ’ πœƒ = βˆ’ 𝑒 2 + s i n c o s C   
  • D πœƒ πœƒ = βˆ’ 𝑒 2 + s i n C   
  • E βˆ’ πœƒ πœƒ βˆ’ πœƒ = 𝑒 + s i n c o s C   

Q5:

Find a relation between 𝑦 and π‘₯, given that π‘₯𝑦𝑦′=π‘₯βˆ’5.

  • A 𝑦 = π‘₯ βˆ’ 1 0 | π‘₯ | +   l n C
  • B 𝑦 = 2 π‘₯ βˆ’ 1 0 π‘₯ +   l n C
  • C 𝑦 = 2 π‘₯ βˆ’ 1 0 | π‘₯ | +   l n C
  • D 𝑦 = π‘₯ βˆ’ 5 | π‘₯ | +   l n C
  • E 𝑦 = π‘₯ 2 βˆ’ 5 | π‘₯ | +   l n C

Q6:

Solve the following differential equation: dd𝑝𝑑=π‘‘π‘βˆ’5𝑝+π‘‘βˆ’5.

  • A 𝑝 = 𝑒 βˆ’ 1 K     
  • B 𝑝 = 𝑒 + 1 K     
  • C 𝑝 = 𝑒 + 1 K       
  • D 𝑝 = 𝑒 βˆ’ 1 K      
  • E 𝑝 = 𝑒 βˆ’ 1 K       

Q7:

Solve the following differential equation: (π‘’βˆ’5)𝑦′=2+π‘₯cos.

  • A 𝑒 βˆ’ 5 𝑦 = 2 π‘₯ βˆ’ π‘₯ +  s i n C
  • B 𝑒 βˆ’ 5 = 2 π‘₯ + π‘₯ +  s i n C
  • C 𝑒 βˆ’ 5 𝑦 = 2 π‘₯ + π‘₯ +  s i n C
  • D 𝑒 βˆ’ 5 𝑦 = π‘₯ + π‘₯ +  s i n C
  • E 𝑒 βˆ’ 5 = 2 π‘₯ βˆ’ π‘₯ +  s i n C

Q8:

Solve the differential equation 𝑦+π‘₯𝑒=0.

  • A 𝑦 = ο€Ύ π‘₯ 2 +  l n C 
  • B 𝑦 = βˆ’ ο€Ή π‘₯ +  l n C 
  • C 𝑦 = βˆ’ ο€Ύ π‘₯ 2 +  l n C 
  • D 𝑦 = ο€Ή 2 π‘₯ +  l n C 
  • E 𝑦 = βˆ’ ο€Ή 2 π‘₯ +  l n C 

Q9:

Solve the differential equation dd𝑦π‘₯=βˆ’5π‘₯βˆšπ‘¦.

  • A √ 𝑦 = βˆ’ 5 π‘₯ 2 +  C or 𝑦=0
  • B √ 𝑦 = βˆ’ 5 π‘₯ +  C or 𝑦=0
  • C 𝑦 = ο€Ύ βˆ’ 5 π‘₯ 4 +    C or 𝑦=0
  • D √ 𝑦 = βˆ’ 5 π‘₯ 4 +  C or 𝑦=0
  • E 𝑦 = ο€Ύ βˆ’ 5 π‘₯ 2 +    C or 𝑦=0

Q10:

Solve the following differential equation by separating it: π‘₯𝑦π‘₯=(1βˆ’π‘¦).dd

  • A 𝑦 = ( | π‘₯ | + ) c o s l n C
  • B 𝑦 = ( | π‘₯ | + ) s i n l n C
  • C 𝑦 = ( | π‘₯ | + ) l n c o s C
  • D 𝑦 = ( | π‘₯ | + ) l n s i n C

Q11:

Find a 1-parameter family of solutions for the differential equation 𝑦𝑦=(𝑦+1), π‘¦β‰ βˆ’1.

  • A 1 𝑦 + 1 βˆ’ | 𝑦 + 1 | = π‘₯ + 𝑐 l n
  • BThere is no solution.
  • C 1 𝑦 + 1 + | 𝑦 + 1 | = π‘₯ + 𝑐 l n
  • D βˆ’ 1 𝑦 + 1 + | 𝑦 + 1 | = π‘₯ + 𝑐 l n

Q12:

Find the implicit solution to the following differential equation: sinddcos(𝑦)𝑦π‘₯βˆ’(π‘₯)=0.

  • A c o s s e c ( 𝑦 ) + ( π‘₯ ) = 𝐢
  • B s i n c o s ( 𝑦 ) + ( π‘₯ ) = 𝐢
  • C c o s c s c ( 𝑦 ) + ( π‘₯ ) = 𝐢
  • D c o s s i n ( 𝑦 ) + ( π‘₯ ) = 𝐢

Q13:

Which of the following is a solution of π‘₯+𝑦𝑦′=0 defined for all βˆ’4<π‘₯<4?

  • A 𝑦 = √ 4 βˆ’ π‘₯ 
  • B 𝑦 = √ 1 6 + π‘₯ 
  • C 𝑦 = √ 4 + π‘₯ 
  • D 𝑦 = √ 1 6 βˆ’ π‘₯ 

Q14:

Find a relation between 𝑒 and 𝑑 given that dd𝑒𝑑=1+𝑑𝑒𝑑+𝑒𝑑οŠͺοŠͺ.

  • A 𝑒 + 𝑒 = 1 𝑑 + 𝑑 3 +    C
  • B 𝑒 + 𝑒 = βˆ’ 1 𝑑 + 𝑑 3 +    C
  • C 𝑒 5 + 𝑒 2 = βˆ’ 1 𝑑 + 𝑑 3 +    C
  • D 𝑒 5 + 𝑒 2 = βˆ’ 1 𝑑 + 𝑑 +    C
  • E 𝑒 5 + 𝑒 2 = 1 𝑑 + 𝑑 3 +    C

Q15:

Solve the differential equation dd𝑧𝑑+𝑒=0οŠ¨οοŠ°οŠ¨ο™.

  • A 𝑧 = βˆ’ 1 2 ο€Ύ 𝑒 2 +  l n C  
  • B 𝑧 = βˆ’ 1 2 ο€Ή 𝑒 +  l n C  
  • C 𝑧 = βˆ’ 1 2 ο€Ή 2 𝑒 +  l n C  
  • D 𝑧 = βˆ’ 1 2 ο€Ή 𝑒 +  l n C  
  • E 𝑧 = 1 2 ο€Ή 𝑒 +  l n C  

Q16:

Solve the differential equation dd𝑦π‘₯+3π‘₯𝑦=6π‘₯.

  • A 𝑦 = 2 βˆ’ 𝑒 C   
  • B 𝑦 = 2 + 𝑒 C   
  • C 𝑦 = 2 π‘₯ 𝑒 + 𝑒        C
  • D 𝑦 = 6 + 𝑒 C   
  • E 𝑦 = 2 π‘₯ 𝑒 +     C

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