Lesson Worksheet: Separable Differential Equations Mathematics • Higher Education

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‘₯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

Q3:

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

Q4:

Solve the following differential equation, using the given boundary conditions to find a particular solution: cossecddcscsec(π‘₯)(𝑦)𝑦π‘₯=(𝑦)(π‘₯), π‘₯=0, 𝑦=πœ‹4.

  • Atantan(𝑦)βˆ’π‘¦=(π‘₯)+1βˆ’πœ‹4
  • Btantan(𝑦)+1βˆ’πœ‹4=(π‘₯)βˆ’π‘₯
  • Ctantan(𝑦)+1βˆ’πœ‹4=(π‘₯)+π‘₯
  • Dtantan(𝑦)+𝑦=(π‘₯)+1βˆ’πœ‹4
  • Etancot(𝑦)βˆ’π‘¦=(π‘₯)+1βˆ’πœ‹4

Q5:

Find a particular solution to the differential equation π‘₯+π‘₯𝑦𝑦π‘₯=π‘₯βˆ’1dd, satisfying the boundary conditions π‘₯=12 and 𝑦=12π‘’οŠ©.

  • A𝑦=π‘₯𝑒
  • B𝑦=12π‘₯π‘’οŽ ο‘
  • C𝑦=π‘₯π‘’οŽ ο‘
  • D𝑦=π‘₯π‘’ο‘οŽ©οŽ ο‘
  • E𝑦=12π‘’ο‘οŽ©οŽ ο‘

Q6:

By expressing 9π‘₯βˆ’17(4π‘₯βˆ’7)(π‘₯βˆ’3) in partial fractions, write down the particular solution to the differential equation (4π‘₯βˆ’7)(π‘₯βˆ’3)𝑦π‘₯=(9π‘₯βˆ’17)𝑦dd, satisfying the condition 𝑦=5 when π‘₯=2. Give your solution in the form 𝑦=𝑓(π‘₯).

  • A𝑦=5(4π‘₯βˆ’7)(π‘₯βˆ’3)
  • B𝑦=(4π‘₯βˆ’7)(π‘₯βˆ’3)
  • C𝑦=5(4π‘₯βˆ’7)(π‘₯βˆ’3)
  • D𝑦=5(4π‘₯βˆ’7)(π‘₯βˆ’3)
  • E𝑦=(4π‘₯βˆ’7)(π‘₯βˆ’3)

Q7:

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

Q8:

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

Q9:

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

Q10:

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

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