Worksheet: Homogeneous First-Order Differential Equations

In this worksheet, we will practice solving first-order homogeneous differential equations by using a substitution to reduce the differential equation to a separable one.

Q1:

Solve the differential equation ๐‘ฅ ( ๐‘ฅ + ๐‘ฆ ) ๐‘ฆ โ€ฒ + ๐‘ฆ ( 3 ๐‘ฅ + ๐‘ฆ ) = 0 .

  • A ๐‘ฆ ๐‘ฅ โˆ’ 2 ๐‘ฆ ๐‘ฅ + ๐ถ = 0 ๏Šจ ๏Šจ ๏Šฉ
  • B ๐‘ฆ + 2 ๐‘ฆ ๐‘ฅ โˆ’ ๐ถ ๐‘ฅ = 0 ๏Šจ ๏Šฉ
  • C ๐‘ฆ โˆ’ 2 ๐‘ฆ ๐‘ฅ + ๐ถ ๐‘ฅ = 0 ๏Šจ ๏Šฉ
  • D ๐‘ฆ ๐‘ฅ + 2 ๐‘ฆ ๐‘ฅ + ๐ถ = 0 ๏Šจ ๏Šจ ๏Šฉ
  • E 2 ๐‘ฆ ๐‘ฅ โˆ’ ๐‘ฆ ๐‘ฅ + ๐ถ = 0 ๏Šฉ ๏Šจ ๏Šจ

Q2:

Solve the differential equation ๐‘ฆ ๐‘ฆ โ€ฒ + ๐‘ฅ = โˆš ๐‘ฅ + ๐‘ฆ ๏Šจ ๏Šจ .

  • A โˆš ๐‘ฅ + ๐‘ฆ = ๐ถ โˆ’ ๐‘ฅ ๏Šจ ๏Šจ
  • B โˆš ๐‘ฆ โˆ’ ๐‘ฅ = ๐‘ฅ + ๐ถ ๏Šจ ๏Šจ
  • C โˆš ๐‘ฅ + ๐‘ฆ = ๐‘ฅ + ๐ถ ๏Šจ ๏Šจ
  • D โˆš ๐‘ฆ โˆ’ ๐‘ฅ = ๐‘ฅ ๐ถ โˆ’ ๐‘ฅ ๏Šจ ๏Šจ
  • E โˆš ๐‘ฅ โˆ’ ๐‘ฆ = ๐‘ฅ + ๐ถ ๏Šจ ๏Šจ

Q3:

Solve the differential equation 2 ๐‘ฅ ๐‘ฆ ๐‘ฆ โ€ฒ = ๐‘ฅ + 2 ๐‘ฆ ๏Šจ ๏Šจ .

  • A ๐‘ฆ = ๐‘ฅ | | | ๐ถ ๐‘ฅ | | | ๏Šจ ๏Šจ l n
  • B 2 ๐‘ฆ + ๐‘ฅ + ๐ถ = 0 ๏Šจ ๏Šจ
  • C ๐‘ฆ = ๐‘ฅ ( | ๐‘ฅ | + ๐ถ ) ๏Šจ ๏Šจ l n
  • D 2 ๐‘ฆ + ๐ถ ๐‘ฅ + ๐‘ฅ = 0 ๏Šจ ๏Šช ๏Šจ
  • E ๐‘ฆ = | ๐ถ ๐‘ฅ | ๏Šจ l n

Q4:

Solve the differential equation ( ๐‘ฅ + ๐‘ฆ ) ๐‘ฆ โ€ฒ = ๐‘ฅ โˆ’ ๐‘ฆ .

  • A ๐‘ฅ โˆ’ 2 ๐‘ฅ ๐‘ฆ โˆ’ ๐‘ฆ = ๐ถ ๏Šจ ๏Šจ
  • B ๐‘ฅ + 2 ๐‘ฅ ๐‘ฆ + ๐‘ฆ = ๐ถ ๐‘ฅ ๏Šจ ๏Šจ ๏Šจ
  • C ๐‘ฅ โˆ’ 2 ๐‘ฅ ๐‘ฆ โˆ’ ๐‘ฆ = ๐ถ ๐‘‹ ๏Šจ ๏Šจ ๏Šช
  • D ๐‘ฅ + 2 ๐‘ฅ ๐‘ฆ โˆ’ ๐‘ฆ = ๐ถ ๏Šจ ๏Šจ
  • E ๐‘ฅ โˆ’ 2 ๐‘ฅ ๐‘ฆ โˆ’ ๐‘ฆ = ๐ถ ๐‘ฅ ๏Šจ ๏Šจ ๏Šฉ

Q5:

Solve the differential equation ๐‘ฅ ๐‘ฆ โ€ฒ = ๐‘ฆ + โˆš ๐‘ฅ + ๐‘ฆ ๏Šจ ๏Šจ .

  • A ๐‘ฆ = ๐‘ฅ ( | ๐ถ ๐‘ฅ | ) c o s l n
  • B ๐‘ฆ = ๐‘ฅ ( | ๐ถ ๐‘ฅ | ) s i n l n
  • C ๐‘ฆ = ๐‘ฅ ( | ๐ถ ๐‘ฅ | ) s i n h l n
  • D ๐‘ฆ = ๐‘ฅ ( | ๐ถ ๐‘ฅ | ) c o s h l n
  • E ๐‘ฆ = ๐‘ฅ ( | ๐ถ ๐‘ฅ | ) t a n l n

Q6:

Find the solution of the differential equation ๐‘ฅ ๐‘ฆ ๐‘ฅ = ๐‘ฆ + โˆš ๐‘ฅ โˆ’ ๐‘ฆ d d ๏Šจ ๏Šจ that satisfies the initial condition ๐‘ฆ ( ๐‘ฅ ) = 0 ๏Šฆ , where ๐‘ฅ > 0 ๏Šฆ .

  • A ๐‘ฆ = ๐‘ฅ ๏€ป ๐‘ฅ ๐‘ฅ ๏‡ s i n l n ๏Šฆ
  • B ๐‘ฆ = ๐‘ฅ ๏€ฝ ๐‘ฅ ๐‘ฅ ๏‰ c o s l n ๏Šฆ
  • C ๐‘ฆ = ๐‘ฅ ๏€ฝ ๐‘ฅ ๐‘ฅ ๏‰ s i n l n ๏Šฆ
  • D ๐‘ฆ = ๐‘ฅ ๏€ป ๐‘ฅ ๐‘ฅ ๏‡ c o s l n ๏Šฆ
  • E ๐‘ฆ = ๏€ป ๐‘ฅ ๐‘ฅ ๏‡ s i n l n ๏Šฆ

Q7:

Solve the differential equation ๐‘ฅ ๐‘ฆ ๐‘ฆ โ€ฒ = ๐‘ฅ + ๐‘ฆ ๏Šจ ๏Šฉ ๏Šฉ .

  • A ๐‘ฆ = 3 ๐‘ฅ โˆš | ๐‘ฅ | + ๐ถ ๏Žข l n
  • B ๐‘ฆ = ๐‘ฅ โˆš | ๐‘ฅ | + ๐ถ ๏Žข l n ๏Šฉ
  • C ๐‘ฆ = ๐‘ฅ โˆš | ๐ถ ๐‘ฅ | ๏Žข l n
  • D ๐‘ฆ = โˆ’ ๐‘ฅ โˆš | ๐ถ ๐‘ฅ | ๏Žข l n ๏Šฉ
  • E ๐‘ฆ = ๐‘ฅ โˆš | ๐ถ ๐‘ฅ | ๏Žข l n ๏Šฉ

Q8:

Solve the differential equation ๐‘ฅ ๐‘ฆ โ€ฒ = ๐‘ฅ ๐‘ฆ + ๐‘ฆ ๏Šจ ๏Šจ .

  • A ๐‘ฆ = ๐‘ฅ | | | ๐ถ ๐‘ฅ | | | l n
  • B ๐‘ฆ = ๐‘ฅ | ๐ถ ๐‘ฅ | l n
  • C ๐‘ฆ = โˆ’ ๐‘ฅ | | | ๐ถ ๐‘ฅ | | | l n
  • D ๐‘ฆ = 1 | | | | l n ๏Œข ๏—
  • E ๐‘ฆ = ๐‘ฅ | | | | l n ๏Œข ๏—

Q9:

Solve the differential equation ๐‘ฅ ๐‘ฆ โ€ฒ = ๐‘ฆ + 2 โˆš ๐‘ฅ ๐‘ฆ .

  • A ๐‘ฆ = ๐‘ฅ | | | ๐ถ ๐‘ฅ | | | l n
  • B โˆš ๐‘ฆ = 2 ๐‘ฅ + ๐ถ โˆš ๐‘ฅ
  • C โˆš ๐‘ฆ = โˆš ๐‘ฅ ( | ๐‘ฅ | + ๐ถ ) l n
  • D โˆš ๐‘ฆ = โˆš ๐‘ฅ | | | ๐ถ ๐‘ฅ | | | l n
  • E โˆš ๐‘ฆ = ๐‘ฅ | | | ๐ถ ๐‘ฅ | | | l n

Q10:

Solve the following differential equation: d d ๐‘ฆ ๐‘ฅ = 2 ๐‘ฅ + 2 ๐‘ฆ ๐‘ฅ ๐‘ฆ . ๏Šช ๏Šช ๏Šฉ

  • A ๐‘ฆ = ๏€น ๐‘ฅ + 2 ๐‘ฅ ๏… C ๏Šช ๏Šฎ ๏Ž  ๏Žฃ
  • B ๐‘ฆ = ๏€น ๐‘ฅ โˆ’ 2 ๐‘ฅ ๏… C ๏Šช ๏Šฎ ๏Ž  ๏Žฃ
  • C ๐‘ฆ = ๏€น ๐‘ฅ + 2 ๐‘ฅ ๏… C ๏Šฎ ๏Šช ๏Ž  ๏Žฃ
  • D ๐‘ฆ = ๏€น ๐‘ฅ โˆ’ 2 ๐‘ฅ ๏… C ๏Šฎ ๏Šช ๏Ž  ๏Žฃ

Q11:

Solve the differential equation ๐‘ฅ ๐‘ฆ ๐‘ฆ โ€ฒ = ๐‘ฆ + ๐‘ฅ โˆš 4 ๐‘ฅ + ๐‘ฆ ๏Šจ ๏Šจ ๏Šจ .

  • A ๐‘ฆ = ๐‘ฅ ( | ๐ถ ๐‘ฅ | + 4 ) ๏Šจ ๏Šจ l n
  • B ๐‘ฆ = ๐‘ฅ ๏€ป โˆš ๐ถ ๐‘‹ + 4 ๏‡ ๏Šจ ๏Šจ ๏Šจ l n
  • C ๐‘ฆ = ๐‘ฅ ๏€ป โˆš ๐ถ ๐‘‹ + 4 ๏‡ ๏Šจ ๏Šจ l n
  • D ๐‘ฆ = ๐‘ฅ ๏€บ | ๐ถ ๐‘ฅ | โˆ’ 4 ๏† ๏Šจ ๏Šจ ๏Šจ l n
  • E ๐‘ฆ = ๐‘ฅ ( | ๐ถ ๐‘ฅ | โˆ’ 4 ) ๏Šจ ๏Šจ l n

Q12:

Solve the differential equation ๐‘ฅ ๐‘ฆ โ€ฒ = ๐‘ฅ ๐‘ฆ + ๐‘ฅ ๐‘’ ๏Šจ ๏Šจ ๏‘‘ ๏‘ .

  • A ๐‘ฆ = โˆ’ ๐‘ฅ | | ๐ถ ๐‘ฅ | | l n l n
  • B ๐‘ฆ = ๐‘ฅ | | ๐ถ ๐‘ฅ | | l n l n
  • C ๐‘ฆ = โˆ’ | | ๐ถ ๐‘ฅ | | l n l n
  • D ๐‘ฆ = ๐‘ฅ | | | | | | ๐ถ ๐‘ฅ | | | | | | l n l n
  • E ๐‘ฆ = โˆ’ ๐‘ฅ | | | | | | ๐ถ ๐‘ฅ | | | | | | l n l n

Q13:

Solve the differential equation ๐‘ฅ ( ๐‘ฅ + ๐‘ฆ ) ๐‘ฆ โ€ฒ = ๐‘ฆ ( ๐‘ฅ โˆ’ ๐‘ฆ ) .

  • A ๐‘ฅ + ๐‘ฆ | ๐‘ฅ ๐‘ฆ | + ๐ถ = 0 l n
  • B ๐‘ฅ โˆ’ ๐‘ฆ | ๐‘ฅ ๐‘ฆ | + ๐ถ = 0 l n
  • C ๐‘ฅ + ๐‘ฆ | ๐‘ฅ ๐‘ฆ | + ๐ถ ๐‘ฆ = 0 l n
  • D ๐‘ฅ โˆ’ ๐‘ฆ | ๐‘ฅ ๐‘ฆ | + ๐ถ ๐‘ฆ = 0 l n
  • E ๐‘ฅ โˆ’ ๐‘ฆ | | | ๐‘ฅ ๐‘ฆ | | | + ๐ถ ๐‘ฆ = 0 l n

Q14:

Solve the differential equation ๐‘ฅ ๐‘ฆ ๐‘ฆ โ€ฒ = ๐‘ฅ + 3 ๐‘ฆ ๏Šจ ๏Šจ .

  • A ๐‘ฆ = ๐ถ ๐‘ฅ + 1 3 ๐‘ฅ ๏Šฎ ๏Šจ
  • B ๐‘ฆ = ๐ถ ๐‘ฅ โˆ’ 1 3 ๐‘ฅ ๏Šจ ๏Šฎ ๏Šจ
  • C ๐‘ฆ = ๐ถ ๐‘ฅ + 1 2 ๐‘ฅ ๏Šจ ๏Šฌ ๏Šจ
  • D ๐‘ฆ = ๐ถ ๐‘ฅ + 1 2 ๐‘ฅ ๏Šจ ๏Šซ
  • E ๐‘ฆ = ๐ถ ๐‘ฅ โˆ’ 1 2 ๐‘ฅ ๏Šจ ๏Šฌ ๏Šจ

Q15:

Solve the differential equation ๏€น ๐‘ฅ โˆ’ ๐‘ฆ ๏… ๐‘ฆ โ€ฒ = 2 ๐‘ฅ ๐‘ฆ ๏Šจ ๏Šจ .

  • A | ๐‘ฆ | = ๐ถ ๏€น ๐‘ฅ โˆ’ ๐‘ฆ ๏… ๏Šจ ๏Šจ
  • B | ๐‘ฆ | = ๐ถ ( ๐‘ฅ + ๐‘ฆ )
  • C | ๐‘ฆ | = ๐ถ ( ๐‘ฅ โˆ’ ๐‘ฆ )
  • D | ๐‘ฆ | = ๐ถ ๏€น ๐‘ฆ โˆ’ ๐‘ฅ ๏… ๏Šจ ๏Šจ
  • E | ๐‘ฆ | = ๐ถ ๏€น ๐‘ฅ + ๐‘ฆ ๏… ๏Šจ ๏Šจ

Q16:

Solve the differential equation ( ๐‘ฅ โˆ’ ๐‘ฆ ) ๐‘ฆ โ€ฒ = ๐‘ฅ + ๐‘ฆ .

  • A 2 ๏€ป ๐‘ฆ ๐‘ฅ ๏‡ + ๏€พ ๐‘ฆ ๐‘ฅ + 1 ๏Š + 2 | ๐‘ฅ | + ๐ถ = 0 t a n l n l n ๏Šฑ ๏Šง ๏Šจ ๏Šจ
  • B 2 ๏€ป ๐‘ฆ ๐‘ฅ ๏‡ โˆ’ ๏€ผ 1 โˆ’ ๏ˆ 2 โˆ’ 2 | ๐‘ฅ | + ๐ถ = 0 t a n h l n l n ๏Šฑ ๏Šง ๏˜ ๏— ๏Žก ๏Žก
  • C 2 ๏€ป ๐‘ฆ ๐‘ฅ ๏‡ + ๏€ผ 1 โˆ’ ๏ˆ 2 โˆ’ 2 | ๐‘ฅ | + ๐ถ = 0 t a n h l n l n ๏Šฑ ๏Šง ๏˜ ๏— ๏Žก ๏Žก
  • D 2 ๏€ป ๐‘ฆ ๐‘ฅ ๏‡ + ๏€พ ๐‘ฆ ๐‘ฅ + 1 ๏Š โˆ’ 2 | ๐‘ฅ | + ๐ถ = 0 t a n l n l n ๏Šฑ ๏Šง ๏Šจ ๏Šจ
  • E 2 ๏€ป ๐‘ฆ ๐‘ฅ ๏‡ โˆ’ ๏€พ ๐‘ฆ ๐‘ฅ + 1 ๏Š โˆ’ 2 | ๐‘ฅ | + ๐ถ = 0 t a n l n l n ๏Šฑ ๏Šง ๏Šจ ๏Šจ

Q17:

Solve the differential equation 2 ๐‘ฅ ๐‘ฆ ๐‘ฆ ๐‘ฅ = 4 ๐‘ฅ + 3 ๐‘ฆ d d ๏Šจ ๏Šจ .

  • A ๐‘ฆ โˆ’ 4 ๐‘ฅ = ๐‘ฅ + ๐‘˜ ๐‘ฅ ๏Šจ ๏Šจ ๏Šฉ ๏Šจ
  • B ๐‘ฆ + 4 ๐‘ฅ = ๐‘˜ ๐‘ฅ ๏Šจ ๏Šจ ๏Šฉ
  • C โˆ’ ๐‘ฆ + 4 ๐‘ฅ = ๐‘˜ ๐‘ฅ ๏Šจ ๏Šจ ๏Šฉ
  • D ๐‘ฆ โˆ’ 4 ๐‘ฅ = ๐‘˜ ๐‘ฅ ๏Šจ ๏Šจ ๏Šฉ
  • E ๐‘ฆ + 4 ๐‘ฅ = ๐‘ฅ + ๐‘˜ ๐‘ฅ ๏Šจ ๏Šจ ๏Šฉ ๏Šจ

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