Worksheet: First-Order Linear Differential Equations

In this worksheet, we will practice solving linear first-order differential equations.

Q1:

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

  • A𝑦=π‘₯βˆ’1+𝑒CοŠ±ο—
  • B𝑦=π‘₯βˆ’1+𝑒C
  • C𝑦=π‘₯𝑒2+π‘’οŠ¨οŠ±ο—οŠ±ο—C
  • D𝑦=π‘₯𝑒2+π‘’οŠ¨ο—οŠ±ο—C
  • E𝑦=π‘₯+1+𝑒CοŠ±ο—

Q2:

Solve the differential equation π‘₯𝑦π‘₯+𝑦=π‘₯π‘₯ddln, where π‘₯>0, subject to the condition 𝑦(1)=0.

  • A𝑦=π‘₯2π‘₯βˆ’π‘₯4+12π‘₯ln
  • B𝑦=π‘₯π‘₯βˆ’π‘₯4+14π‘₯ln
  • C𝑦=π‘₯4π‘₯βˆ’π‘₯4+14π‘₯ln
  • D𝑦=π‘₯2π‘₯+π‘₯4+14π‘₯ln
  • E𝑦=π‘₯2π‘₯βˆ’π‘₯4+14π‘₯ln

Q3:

Solve the differential equation π‘₯𝑦π‘₯+π‘₯𝑦=1dd, where π‘₯>0, subject to the condition 𝑦(1)=2.

  • A𝑦=π‘₯+2π‘₯ln
  • B𝑦=π‘₯βˆ’2+2π‘₯lnln
  • C𝑦=π‘₯+22π‘₯ln
  • D𝑦=π‘₯π‘₯ln
  • E𝑦=βˆ’1π‘₯+3

Q4:

Solve the differential equation π‘₯𝑦π‘₯=𝑦+π‘₯π‘₯ddsin subject to the condition 𝑦(πœ‹)=0.

  • A𝑦=π‘₯π‘₯+π‘₯cos
  • B𝑦=π‘₯π‘₯βˆ’π‘₯cos
  • C𝑦=βˆ’π‘₯π‘₯+π‘₯cos
  • D𝑦=π‘₯π‘₯cos
  • E𝑦=βˆ’π‘₯π‘₯βˆ’π‘₯cos

Q5:

Solve the differential equation 𝑑𝑒𝑑=𝑑+3𝑒dd subject to the condition 𝑒(2)=4.

  • A𝑒=π‘‘οŠ¨
  • B𝑒=βˆ’π‘‘+π‘‘οŠ©
  • C𝑒=𝑑5+1285π‘‘οŠ¨οŠ©
  • D𝑒=βˆ’π‘‘+π‘‘οŠ¨οŠ©
  • E𝑒=βˆ’π‘‘βˆ’π‘‘οŠ¨οŠ©

Q6:

Solve the differential equation 𝑑𝑦𝑑+3𝑑𝑦=√1+π‘‘οŠ¨οŠ¨dd, where 𝑑>0.

  • A𝑦=13ο€Ή1+𝑑𝑑+π‘‘οŠ¨οŠ±οŠ©οŠ±οŠ©οŽ’οŽ‘C
  • B𝑦=13(1+𝑑)𝑑+π‘‘οŽ’οŽ‘οŠ±οŠ©οŠ±οŠ©C
  • C𝑦=13(1+𝑑)𝑑+π‘‘οŽ’οŽ‘οŠ±οŠ©οŠ±οŠ©C
  • D𝑦=13ο€Ή1+𝑑𝑑+π‘‘οŠ¨οŠ±οŠ©οŠ±οŠ©οŽ οŽ‘C
  • E𝑦=ο€Ή1+𝑑𝑑+π‘‘οŠ¨οŠ±οŠ©οŠ±οŠ©οŽ’οŽ‘C

Q7:

Solve the differential equation π‘₯𝑦π‘₯+𝑦=√π‘₯dd.

  • A𝑦=√π‘₯2+π‘₯C
  • B𝑦=25√π‘₯+π‘₯C
  • C𝑦=2√π‘₯3+π‘₯C
  • D𝑦=25√π‘₯+π‘₯C
  • E𝑦=2√π‘₯3+C

Q8:

Solve the differential equation 2π‘₯𝑦π‘₯+𝑦=2√π‘₯dd.

  • A𝑦=√π‘₯+√π‘₯C
  • B𝑦=√π‘₯+√π‘₯C
  • C𝑦=π‘₯+√π‘₯C
  • D𝑦=1+√π‘₯C
  • E𝑦=π‘₯+π‘₯C

Q9:

Solve the differential equation ο€Ήπ‘₯+1𝑦π‘₯+3π‘₯(π‘¦βˆ’1)=0dd subject to the condition 𝑦(0)=2.

  • A𝑦=1+1(π‘₯+1)
  • B𝑦=1βˆ’1(π‘₯+1)
  • C𝑦=1+1(π‘₯+1)
  • D𝑦=1βˆ’5√5(π‘₯+1)
  • E𝑦=3+1(π‘₯+1)

Q10:

Is the differential equation ddcosπ‘Ÿπ‘‘+π‘‘π‘Ÿ=π‘’οŠ±ο linear?

  • Ayes
  • Bno

Q11:

Solve the differential equation π‘₯𝑦π‘₯βˆ’2𝑦=π‘₯dd, where π‘₯>0.

  • A𝑦=βˆ’π‘₯(π‘₯+)lnC
  • B𝑦=π‘₯(π‘₯+)lnC
  • C𝑦=π‘₯(π‘₯+)C
  • D14π‘₯+π‘₯C
  • E𝑦=βˆ’12π‘₯(π‘₯+)lnC

Q12:

Solve the differential equation π‘₯𝑦π‘₯+2π‘₯𝑦=π‘₯ddln subject to the condition 𝑦(1)=2.

  • A𝑦=π‘₯βˆ’π‘₯+3π‘₯ln
  • B𝑦=π‘₯π‘₯βˆ’π‘₯+3π‘₯ln
  • C𝑦=π‘₯π‘₯βˆ’π‘₯+1π‘₯ln
  • D𝑦=π‘₯π‘₯+3π‘₯ln
  • E𝑦=π‘₯+42π‘₯ln

Q13:

Solve the differential equation 4π‘₯𝑦+π‘₯𝑦π‘₯=π‘₯οŠͺddsin.

  • A𝑦=π‘₯βˆ’π‘₯+3π‘₯coscosCοŠͺ
  • B𝑦=βˆ’π‘₯+3π‘₯+3π‘₯coscosCοŠͺ
  • C𝑦=π‘₯+3π‘₯+3π‘₯coscosCοŠͺ
  • D𝑦=βˆ’π‘₯+3π‘₯+3π‘₯coscosCοŠͺ
  • E𝑦=π‘₯βˆ’3π‘₯+3π‘₯coscosCοŠͺ

Q14:

Solve the differential equation 𝑑𝑦𝑑+3𝑑𝑦=π‘‘οŠ©οŠ¨ddcos subject to the condition 𝑦(πœ‹)=0.

  • A𝑦=𝑑+1𝑑sin
  • B𝑦=𝑑𝑑sin
  • C𝑦=βˆ’π‘‘+1𝑑sin
  • D𝑦=βˆ’π‘‘π‘‘sin
  • E𝑦=𝑑𝑑+𝑑+1𝑑sincos

Q15:

Is the differential equation 𝑒𝑒=𝑑+βˆšπ‘‘π‘’π‘‘οdd linear?

  • Ayes
  • Bno

Q16:

Solve the differential equation π‘‘π‘‘π‘Ÿπ‘‘+π‘Ÿ=𝑑𝑒lndd, where 𝑑>0.

  • Aπ‘Ÿ=𝑒+2π‘‘οŠ±οCln
  • Bπ‘Ÿ=𝑒+𝑑Cln
  • Cπ‘Ÿ=𝑒+π‘‘οŠ±οCln
  • Dπ‘Ÿ=𝑒+2𝑑Cln
  • Eπ‘Ÿ=1+𝑑Cln

Q17:

Solve the differential equation dd𝑦π‘₯βˆ’π‘¦=𝑒.

  • A𝑦=βˆ’(π‘₯+)𝑒C
  • B𝑦=(π‘₯+)𝑒C
  • C𝑦=(π‘₯+)𝑒CοŠ¨ο—
  • D𝑦=(π‘₯+)𝑒CοŠ±ο—
  • E𝑦=(𝑒+)𝑒C

Q18:

Is the differential equation ddtan𝑦π‘₯βˆ’π‘₯=𝑦π‘₯ linear?

  • Ano
  • Byes

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