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‘ฆโ€ฒ+๐‘ฆ(3๐‘ฅ+๐‘ฆ)=0.

  • A2๐‘ฆ๐‘ฅโˆ’๐‘ฆ๐‘ฅ+๐ถ=0๏Šฉ๏Šจ๏Šจ
  • B๐‘ฆ+2๐‘ฆ๐‘ฅโˆ’๐ถ๐‘ฅ=0๏Šจ๏Šฉ
  • C๐‘ฆ๐‘ฅ+2๐‘ฆ๐‘ฅ+๐ถ=0๏Šจ๏Šจ๏Šฉ
  • D๐‘ฆ๐‘ฅโˆ’2๐‘ฆ๐‘ฅ+๐ถ=0๏Šจ๏Šจ๏Šฉ
  • E๐‘ฆโˆ’2๐‘ฆ๐‘ฅ+๐ถ๐‘ฅ=0๏Šจ๏Šฉ

Q2:

Solve the differential equation ๐‘ฆ๐‘ฆโ€ฒ+๐‘ฅ=โˆš๐‘ฅ+๐‘ฆ๏Šจ๏Šจ.

  • Aโˆš๐‘ฆโˆ’๐‘ฅ=๐‘ฅ+๐ถ๏Šจ๏Šจ
  • Bโˆš๐‘ฅโˆ’๐‘ฆ=๐‘ฅ+๐ถ๏Šจ๏Šจ
  • Cโˆš๐‘ฅ+๐‘ฆ=๐‘ฅ+๐ถ๏Šจ๏Šจ
  • Dโˆš๐‘ฅ+๐‘ฆ=๐ถโˆ’๐‘ฅ๏Šจ๏Šจ
  • Eโˆš๐‘ฆโˆ’๐‘ฅ=๐‘ฅ๐ถโˆ’๐‘ฅ๏Šจ๏Šจ

Q3:

Solve the differential equation 2๐‘ฅ๐‘ฆ๐‘ฆโ€ฒ=๐‘ฅ+2๐‘ฆ๏Šจ๏Šจ.

  • A๐‘ฆ=๐‘ฅ(|๐‘ฅ|+๐ถ)๏Šจ๏Šจln
  • B2๐‘ฆ+๐‘ฅ+๐ถ=0๏Šจ๏Šจ
  • C๐‘ฆ=|๐ถ๐‘ฅ|๏Šจln
  • D2๐‘ฆ+๐ถ๐‘ฅ+๐‘ฅ=0๏Šจ๏Šช๏Šจ
  • E๐‘ฆ=๐‘ฅ|||๐ถ๐‘ฅ|||๏Šจ๏Šจln

Q4:

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

  • A๐‘ฅโˆ’2๐‘ฅ๐‘ฆโˆ’๐‘ฆ=๐ถ๐‘ฅ๏Šจ๏Šจ๏Šฉ
  • B๐‘ฅโˆ’2๐‘ฅ๐‘ฆโˆ’๐‘ฆ=๐ถ๐‘‹๏Šจ๏Šจ๏Šช
  • C๐‘ฅโˆ’2๐‘ฅ๐‘ฆโˆ’๐‘ฆ=๐ถ๏Šจ๏Šจ
  • D๐‘ฅ+2๐‘ฅ๐‘ฆ+๐‘ฆ=๐ถ๐‘ฅ๏Šจ๏Šจ๏Šจ
  • E๐‘ฅ+2๐‘ฅ๐‘ฆโˆ’๐‘ฆ=๐ถ๏Šจ๏Šจ

Q5:

Solve the differential equation ๐‘ฅ๐‘ฆโ€ฒ=๐‘ฆ+โˆš๐‘ฅ+๐‘ฆ๏Šจ๏Šจ.

  • A๐‘ฆ=๐‘ฅ(|๐ถ๐‘ฅ|)coshln
  • B๐‘ฆ=๐‘ฅ(|๐ถ๐‘ฅ|)tanln
  • C๐‘ฆ=๐‘ฅ(|๐ถ๐‘ฅ|)sinln
  • D๐‘ฆ=๐‘ฅ(|๐ถ๐‘ฅ|)cosln
  • E๐‘ฆ=๐‘ฅ(|๐ถ๐‘ฅ|)sinhln

Q6:

Find the solution of the differential equation ๐‘ฅ๐‘ฆ๐‘ฅ=๐‘ฆ+โˆš๐‘ฅโˆ’๐‘ฆdd๏Šจ๏Šจ that satisfies the initial condition ๐‘ฆ(๐‘ฅ)=0๏Šฆ, where ๐‘ฅ>0๏Šฆ.

  • A๐‘ฆ=๏€ป๐‘ฅ๐‘ฅ๏‡sinln๏Šฆ
  • B๐‘ฆ=๐‘ฅ๏€ฝ๐‘ฅ๐‘ฅ๏‰sinln๏Šฆ
  • C๐‘ฆ=๐‘ฅ๏€ฝ๐‘ฅ๐‘ฅ๏‰cosln๏Šฆ
  • D๐‘ฆ=๐‘ฅ๏€ป๐‘ฅ๐‘ฅ๏‡sinln๏Šฆ
  • E๐‘ฆ=๐‘ฅ๏€ป๐‘ฅ๐‘ฅ๏‡cosln๏Šฆ

Q7:

Solve the differential equation ๐‘ฅ๐‘ฆ๐‘ฆโ€ฒ=๐‘ฅ+๐‘ฆ๏Šจ๏Šฉ๏Šฉ.

  • A๐‘ฆ=๐‘ฅโˆš|๐ถ๐‘ฅ|๏Žขln
  • B๐‘ฆ=3๐‘ฅโˆš|๐‘ฅ|+๐ถ๏Žขln
  • C๐‘ฆ=๐‘ฅโˆš|๐‘ฅ|+๐ถ๏Žขln๏Šฉ
  • D๐‘ฆ=๐‘ฅโˆš|๐ถ๐‘ฅ|๏Žขln๏Šฉ
  • E๐‘ฆ=โˆ’๐‘ฅโˆš|๐ถ๐‘ฅ|๏Žขln๏Šฉ

Q8:

Solve the differential equation ๐‘ฅ๐‘ฆโ€ฒ=๐‘ฅ๐‘ฆ+๐‘ฆ๏Šจ๏Šจ.

  • A๐‘ฆ=๐‘ฅ||||ln๏Œข๏—
  • B๐‘ฆ=๐‘ฅ|||๐ถ๐‘ฅ|||ln
  • C๐‘ฆ=โˆ’๐‘ฅ|||๐ถ๐‘ฅ|||ln
  • D๐‘ฆ=๐‘ฅ|๐ถ๐‘ฅ|ln
  • E๐‘ฆ=1||||ln๏Œข๏—

Q9:

Solve the differential equation ๐‘ฅ๐‘ฆโ€ฒ=๐‘ฆ+2โˆš๐‘ฅ๐‘ฆ.

  • Aโˆš๐‘ฆ=โˆš๐‘ฅ(|๐‘ฅ|+๐ถ)ln
  • Bโˆš๐‘ฆ=๐‘ฅ|||๐ถ๐‘ฅ|||ln
  • Cโˆš๐‘ฆ=2๐‘ฅ+๐ถโˆš๐‘ฅ
  • Dโˆš๐‘ฆ=โˆš๐‘ฅ|||๐ถ๐‘ฅ|||ln
  • E๐‘ฆ=๐‘ฅ|||๐ถ๐‘ฅ|||ln

Q10:

Solve the following differential equation: dd๐‘ฆ๐‘ฅ=2๐‘ฅ+2๐‘ฆ๐‘ฅ๐‘ฆ.๏Šช๏Šช๏Šฉ

  • A๐‘ฆ=๏€น๐‘ฅโˆ’2๐‘ฅ๏…C๏Šช๏Šฎ๏Ž ๏Žฃ
  • B๐‘ฆ=๏€น๐‘ฅ+2๐‘ฅ๏…C๏Šช๏Šฎ๏Ž ๏Žฃ
  • C๐‘ฆ=๏€น๐‘ฅโˆ’2๐‘ฅ๏…C๏Šฎ๏Šช๏Ž ๏Žฃ
  • D๐‘ฆ=๏€น๐‘ฅ+2๐‘ฅ๏…C๏Šฎ๏Šช๏Ž ๏Žฃ

Q11:

Solve the differential equation ๐‘ฅ๐‘ฆ๐‘ฆโ€ฒ=๐‘ฆ+๐‘ฅโˆš4๐‘ฅ+๐‘ฆ๏Šจ๏Šจ๏Šจ.

  • A๐‘ฆ=๐‘ฅ๏€บ|๐ถ๐‘ฅ|โˆ’4๏†๏Šจ๏Šจ๏Šจln
  • B๐‘ฆ=๐‘ฅ(|๐ถ๐‘ฅ|+4)๏Šจ๏Šจln
  • C๐‘ฆ=๐‘ฅ๏€ปโˆš๐ถ๐‘‹+4๏‡๏Šจ๏Šจ๏Šจln
  • D๐‘ฆ=๐‘ฅ(|๐ถ๐‘ฅ|โˆ’4)๏Šจ๏Šจln
  • E๐‘ฆ=๐‘ฅ๏€ปโˆš๐ถ๐‘‹+4๏‡๏Šจ๏Šจln

Q12:

Solve the differential equation ๐‘ฅ๐‘ฆโ€ฒ=๐‘ฅ๐‘ฆ+๐‘ฅ๐‘’๏Šจ๏Šจ๏‘‘๏‘.

  • A๐‘ฆ=โˆ’||๐ถ๐‘ฅ||lnln
  • B๐‘ฆ=๐‘ฅ||๐ถ๐‘ฅ||lnln
  • C๐‘ฆ=๐‘ฅ||||||๐ถ๐‘ฅ||||||lnln
  • D๐‘ฆ=โˆ’๐‘ฅ||๐ถ๐‘ฅ||lnln
  • E๐‘ฆ=โˆ’๐‘ฅ||||||๐ถ๐‘ฅ||||||lnln

Q13:

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

  • A๐‘ฅ+๐‘ฆ|๐‘ฅ๐‘ฆ|+๐ถ=0ln
  • B๐‘ฅ+๐‘ฆ|๐‘ฅ๐‘ฆ|+๐ถ๐‘ฆ=0ln
  • C๐‘ฅโˆ’๐‘ฆ|๐‘ฅ๐‘ฆ|+๐ถ=0ln
  • D๐‘ฅโˆ’๐‘ฆ|||๐‘ฅ๐‘ฆ|||+๐ถ๐‘ฆ=0ln
  • E๐‘ฅโˆ’๐‘ฆ|๐‘ฅ๐‘ฆ|+๐ถ๐‘ฆ=0ln

Q14:

Solve the differential equation ๐‘ฅ๐‘ฆ๐‘ฆโ€ฒ=๐‘ฅ+3๐‘ฆ๏Šจ๏Šจ.

  • A๐‘ฆ=๐ถ๐‘ฅ+12๐‘ฅ๏Šจ๏Šซ
  • B๐‘ฆ=๐ถ๐‘ฅ+13๐‘ฅ๏Šฎ๏Šจ
  • C๐‘ฆ=๐ถ๐‘ฅโˆ’12๐‘ฅ๏Šจ๏Šฌ๏Šจ
  • D๐‘ฆ=๐ถ๐‘ฅ+12๐‘ฅ๏Šจ๏Šฌ๏Šจ
  • E๐‘ฆ=๐ถ๐‘ฅโˆ’13๐‘ฅ๏Šจ๏Šฎ๏Šจ

Q15:

Solve the differential equation ๏€น๐‘ฅโˆ’๐‘ฆ๏…๐‘ฆโ€ฒ=2๐‘ฅ๐‘ฆ๏Šจ๏Šจ.

  • A|๐‘ฆ|=๐ถ๏€น๐‘ฅโˆ’๐‘ฆ๏…๏Šจ๏Šจ
  • B|๐‘ฆ|=๐ถ๏€น๐‘ฆโˆ’๐‘ฅ๏…๏Šจ๏Šจ
  • C|๐‘ฆ|=๐ถ๏€น๐‘ฅ+๐‘ฆ๏…๏Šจ๏Šจ
  • D|๐‘ฆ|=๐ถ(๐‘ฅโˆ’๐‘ฆ)
  • E|๐‘ฆ|=๐ถ(๐‘ฅ+๐‘ฆ)

Q16:

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

  • A2๏€ป๐‘ฆ๐‘ฅ๏‡+๏€พ๐‘ฆ๐‘ฅ+1๏Šโˆ’2|๐‘ฅ|+๐ถ=0tanlnln๏Šฑ๏Šง๏Šจ๏Šจ
  • B2๏€ป๐‘ฆ๐‘ฅ๏‡โˆ’๏€ผ1โˆ’๏ˆ2โˆ’2|๐‘ฅ|+๐ถ=0tanhlnln๏Šฑ๏Šง๏˜๏—๏Žก๏Žก
  • C2๏€ป๐‘ฆ๐‘ฅ๏‡+๏€พ๐‘ฆ๐‘ฅ+1๏Š+2|๐‘ฅ|+๐ถ=0tanlnln๏Šฑ๏Šง๏Šจ๏Šจ
  • D2๏€ป๐‘ฆ๐‘ฅ๏‡โˆ’๏€พ๐‘ฆ๐‘ฅ+1๏Šโˆ’2|๐‘ฅ|+๐ถ=0tanlnln๏Šฑ๏Šง๏Šจ๏Šจ
  • E2๏€ป๐‘ฆ๐‘ฅ๏‡+๏€ผ1โˆ’๏ˆ2โˆ’2|๐‘ฅ|+๐ถ=0tanhlnln๏Šฑ๏Šง๏˜๏—๏Žก๏Žก

Q17:

Solve the differential equation 2๐‘ฅ๐‘ฆ๐‘ฆ๐‘ฅ=4๐‘ฅ+3๐‘ฆdd๏Šจ๏Šจ.

  • Aโˆ’๐‘ฆ+4๐‘ฅ=๐‘˜๐‘ฅ๏Šจ๏Šจ๏Šฉ
  • B๐‘ฆ+4๐‘ฅ=๐‘ฅ+๐‘˜๐‘ฅ๏Šจ๏Šจ๏Šฉ๏Šจ
  • C๐‘ฆโˆ’4๐‘ฅ=๐‘˜๐‘ฅ๏Šจ๏Šจ๏Šฉ
  • D๐‘ฆ+4๐‘ฅ=๐‘˜๐‘ฅ๏Šจ๏Šจ๏Šฉ
  • E๐‘ฆโˆ’4๐‘ฅ=๐‘ฅ+๐‘˜๐‘ฅ๏Šจ๏Šจ๏Šฉ๏Šจ

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