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.

  • A𝑦=π‘₯𝑒+π‘’οŠ±ο—οŠ±ο—C
  • B𝑦=1+𝑒C
  • C𝑦=1+𝑒CοŠ±ο—
  • D𝑦=π‘₯+𝑒CοŠ±ο—
  • E𝑦=π‘₯+𝑒C

Q2:

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

  • A𝑦=115π‘₯+C or 𝑦=0
  • B𝑦=βˆ’115π‘₯+C or 𝑦=0
  • C𝑦=βˆ’35π‘₯+C or 𝑦=0
  • D𝑦=15π‘₯+C or 𝑦=0
  • E𝑦=35π‘₯+C or 𝑦=0

Q3:

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

  • Aβˆ’π»π»βˆ’1𝐻=23ο€Ή1+𝑅+lnC
  • Bβˆ’π»π»βˆ’1𝐻=12ο€Ή1+𝑅+lnC
  • ClnC𝐻𝐻+1𝐻=13ο€Ή1+𝑅+
  • Dβˆ’π»π»βˆ’1𝐻=13ο€Ή1+𝑅+lnC
  • Eβˆ’π»π»+1𝐻=13ο€Ή1+𝑅+lnC

Q4:

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

  • Aπœƒπœƒ+πœƒ=𝑒+sincosC
  • Bπœƒπœƒ+πœƒ=βˆ’π‘’2+sincosC
  • Cβˆ’πœƒπœƒβˆ’πœƒ=βˆ’π‘’2+sincosC
  • Dπœƒπœƒ=βˆ’π‘’2+sinC
  • Eβˆ’πœƒπœƒβˆ’πœƒ=𝑒+sincosC

Q5:

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

  • A𝑦=π‘₯βˆ’10|π‘₯|+lnC
  • B𝑦=2π‘₯βˆ’10π‘₯+lnC
  • C𝑦=2π‘₯βˆ’10|π‘₯|+lnC
  • D𝑦=π‘₯βˆ’5|π‘₯|+lnC
  • E𝑦=π‘₯2βˆ’5|π‘₯|+lnC

Q6:

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

  • A𝑝=π‘’βˆ’1K
  • B𝑝=𝑒+1K
  • C𝑝=𝑒+1K
  • D𝑝=π‘’βˆ’1K
  • E𝑝=π‘’βˆ’1K

Q7:

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

  • Aπ‘’βˆ’5𝑦=2π‘₯βˆ’π‘₯+sinC
  • Bπ‘’βˆ’5=2π‘₯+π‘₯+sinC
  • Cπ‘’βˆ’5𝑦=2π‘₯+π‘₯+sinC
  • Dπ‘’βˆ’5𝑦=π‘₯+π‘₯+sinC
  • Eπ‘’βˆ’5=2π‘₯βˆ’π‘₯+sinC

Q8:

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

  • A𝑦=ο€Ύπ‘₯2+lnC
  • B𝑦=βˆ’ο€Ήπ‘₯+lnC
  • C𝑦=βˆ’ο€Ύπ‘₯2+lnC
  • D𝑦=ο€Ή2π‘₯+lnC
  • E𝑦=βˆ’ο€Ή2π‘₯+lnC

Q9:

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

  • Aβˆšπ‘¦=βˆ’5π‘₯2+C or 𝑦=0
  • Bβˆšπ‘¦=βˆ’5π‘₯+C or 𝑦=0
  • C𝑦=ο€Ύβˆ’5π‘₯4+C or 𝑦=0
  • Dβˆšπ‘¦=βˆ’5π‘₯4+C or 𝑦=0
  • E𝑦=ο€Ύβˆ’5π‘₯2+C or 𝑦=0

Q10:

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

  • A𝑦=(|π‘₯|+)coslnC
  • B𝑦=(|π‘₯|+)sinlnC
  • C𝑦=(|π‘₯|+)lncosC
  • D𝑦=(|π‘₯|+)lnsinC

Q11:

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

  • A1𝑦+1βˆ’|𝑦+1|=π‘₯+𝑐ln
  • BThere is no solution.
  • C1𝑦+1+|𝑦+1|=π‘₯+𝑐ln
  • Dβˆ’1𝑦+1+|𝑦+1|=π‘₯+𝑐ln

Q12:

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

  • Acossec(𝑦)+(π‘₯)=𝐢
  • Bsincos(𝑦)+(π‘₯)=𝐢
  • Ccoscsc(𝑦)+(π‘₯)=𝐢
  • Dcossin(𝑦)+(π‘₯)=𝐢

Q13:

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

  • A𝑦=√4βˆ’π‘₯
  • B𝑦=√16+π‘₯
  • C𝑦=√4+π‘₯
  • D𝑦=√16βˆ’π‘₯

Q14:

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

  • A𝑒+𝑒=1𝑑+𝑑3+C
  • B𝑒+𝑒=βˆ’1𝑑+𝑑3+C
  • C𝑒5+𝑒2=βˆ’1𝑑+𝑑3+C
  • D𝑒5+𝑒2=βˆ’1𝑑+𝑑+C
  • E𝑒5+𝑒2=1𝑑+𝑑3+C

Q15:

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

  • A𝑧=βˆ’12𝑒2+lnC
  • B𝑧=βˆ’12𝑒+lnC
  • C𝑧=βˆ’12ο€Ή2𝑒+lnC
  • D𝑧=βˆ’12𝑒+lnC
  • E𝑧=12𝑒+lnC

Q16:

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

  • A𝑦=2βˆ’π‘’CοŠ±ο—οŽ’
  • B𝑦=2+𝑒CοŠ±ο—οŽ’
  • C𝑦=2π‘₯𝑒+π‘’οŠ©οŠ±ο—οŠ±ο—οŽ’οŽ’C
  • D𝑦=6+𝑒CοŠ±ο—οŽ’
  • E𝑦=2π‘₯𝑒+οŠ©οŠ±ο—οŽ’C

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