Worksheet: Exact Differential Equations

In this worksheet, we will practice identifying and solving first-order exact differential equations.

Q1:

Is the differential equation 𝑥 + 𝑦 𝑥 + 𝑥 + 𝑦 1 + 3 𝑦 𝑦 = 0 t a n d d exact?

  • A yes
  • B no

Q2:

Solve the exact differential equation 2 𝑥 𝑦 + 3 𝑥 𝑥 + 2 𝑥 𝑦 + 4 𝑦 𝑦 = 0 d d .

  • A 2 𝑥 𝑦 + 𝑥 + 𝑦 2 3 𝑥 𝑦 3 𝑥 𝑦 = C
  • B 𝑥 𝑦 + 𝑥 = C
  • C 𝑥 𝑦 + 𝑥 + 4 𝑦 = C
  • D 𝑥 𝑦 + 𝑥 + 2 𝑥 𝑦 + 4 𝑦 2 𝑥 𝑦 = C
  • E 𝑥 𝑦 + 𝑥 + 𝑦 = C

Q3:

Solve the exact differential equation ( 1 + 𝑦 𝑒 ) 𝑥 + ( 2 𝑦 + 𝑥 𝑒 ) 𝑦 = 0 d d .

  • A 𝑥 + 𝑒 + 2 𝑦 = C
  • B 𝑥 + 𝑦 𝑦 + 2 + 1 𝑥 𝑒 = C
  • C 𝑥 + 𝑒 = C
  • D 𝑥 + 2 𝑦 + ( 2 + 𝑥 ) 𝑒 = C
  • E 𝑥 + 𝑒 + 𝑦 = C

Q4:

Is the differential equation 2 𝑥 3 𝑦 2 𝑥 𝑦 𝑥 + 3 𝑦 2 𝑥 3 𝑥 𝑦 𝑦 = 0 d d exact?

  • A no
  • B yes

Q5:

Is the differential equation 𝑥 + 𝑦 𝑥 𝑥 + 𝑦 + 5 𝑥 𝑦 = 0 d l n d exact?

  • A no
  • B yes

Q6:

Solve the exact differential equation ( 𝑥 + 𝑦 ) 𝑥 + 𝑥 𝑦 + 𝑒 𝑦 = 0 c o s l n d d .

  • A 𝑥 𝑥 𝑦 + 𝑒 = s i n l n C
  • B s i n l n C 𝑥 + 𝑥 𝑦 + 𝑒 =
  • C s i n l n C 𝑥 + 𝑥 𝑦 =
  • D 𝑥 + 𝑥 𝑦 = s i n l n C
  • E 𝑥 + 𝑥 𝑦 + 𝑒 = s i n l n C

Q7:

Solve the exact differential equation 2 𝑥 3 𝑦 2 𝑥 𝑦 𝑥 + 3 𝑦 2 𝑥 3 𝑥 𝑦 𝑦 = 0 d d .

  • A 𝑥 𝑦 + 7 2 𝑦 𝑥 = C
  • B 2 3 𝑦 + 𝑥 = C
  • C 𝑦 + 𝑦 𝑥 = C
  • D 2 3 𝑥 𝑦 + 𝑥 = C
  • E 𝑥 𝑦 + 𝑦 𝑥 = C

Q8:

Solve the exact differential equation 𝑥 + 𝑦 𝑥 𝑥 + 𝑦 + 𝑥 𝑦 = 0 d l n d .

  • A 𝑥 4 + 𝑦 𝑥 + 𝑦 3 = l n C
  • B 𝑥 4 + 𝑦 𝑥 + 𝑦 = l n C
  • C 𝑥 4 + 𝑦 𝑥 = l n C
  • D 𝑥 4 + 𝑦 𝑥 + 𝑦 𝑥 = l n C
  • E 𝑥 4 + 2 𝑦 𝑥 + 𝑦 3 𝑥 𝑦 𝑦 2 𝑥 = l n C

Q9:

Solve the exact differential equation 𝑥 + 𝑦 𝑥 + 𝑥 + 𝑦 1 + 𝑦 𝑦 = 0 t a n d d .

  • A 𝑥 2 + 𝑥 𝑦 + 𝑦 1 + 𝑦 = t a n C
  • B 𝑥 2 + 𝑥 𝑦 = t a n C
  • C 2 𝑥 + 𝑥 𝑦 + 𝑦 1 + 𝑦 = t a n C
  • D 𝑥 2 + 𝑥 𝑦 + 1 + 𝑦 2 = t a n l n C
  • E 𝑥 2 + 𝑥 𝑦 + 1 + 𝑦 = t a n l n C

Q10:

Is the differential equation 3 𝑥 + 2 𝑦 𝑥 + 4 𝑥 𝑦 + 6 𝑦 𝑦 = 0 d d exact?

  • A No
  • B Yes

Q11:

Find the general solution for the following exact differential equation: 1 𝑦 𝑥 𝑦 𝑦 𝑥 = 0 . d d

  • A 𝑦 = 𝑥 + 𝑐
  • B 𝑦 = 𝑐 𝑥
  • C 𝑦 = 𝑐 𝑥
  • D 𝑦 = 𝑥 𝑐

Q12:

Solve the exact differential equation 6 𝑥 𝑦 𝑦 𝑥 + 4 𝑦 + 3 𝑥 3 𝑥 𝑦 𝑦 = 0 d d .

  • A 4 𝑥 𝑦 + 𝑥 3 2 𝑥 𝑦 + 4 𝑦 = C
  • B 3 𝑥 𝑦 𝑥 𝑦 = C
  • C 4 𝑥 𝑦 + 𝑥 3 2 𝑥 𝑦 + 2 𝑦 = C
  • D 3 𝑥 𝑦 𝑥 𝑦 + 2 𝑦 = C
  • E 3 𝑥 𝑦 𝑥 𝑦 + 4 𝑦 = C

Q13:

Solve the exact differential equation ( 2 𝑥 + 3 𝑦 ) 𝑥 + ( 3 𝑥 + 2 𝑦 ) 𝑦 = 0 d d .

  • A 𝑥 + 3 𝑥 𝑦 + 2 𝑦 = C
  • B 𝑥 + 4 𝑥 𝑦 1 2 𝑦 = C
  • C 𝑥 + 3 𝑥 𝑦 = C
  • D 𝑥 + 3 𝑥 𝑦 + 𝑦 = C
  • E 𝑥 + 3 𝑥 𝑦 + 𝑥 𝑦 = C

Q14:

Is the differential equation ( 2 𝑥 + 3 𝑦 ) 𝑥 + ( 3 𝑥 + 2 𝑦 ) 𝑦 = 0 d d exact?

  • A no
  • B yes

Q15:

Solve the exact differential equation 𝑦 𝑥 + 3 𝑥 𝑦 𝑦 = 0 d d .

  • A 𝑦 = C
  • B 3 2 𝑥 𝑦 = C
  • C 𝑥 𝑦 = C
  • D 2 𝑥 𝑦 𝑦 4 = C
  • E 𝑥 𝑦 + 3 2 𝑥 𝑦 = C

Q16:

Solve the exact differential equation ( 𝑒 𝑦 + 𝑦 ) 𝑥 + 𝑒 𝑦 + 𝑥 𝑦 𝑦 = 0 s i n t a n d c o s s e c d .

  • A 𝑒 𝑦 + 𝑥 2 𝑦 2 𝑒 𝑦 = c o s s e c s i n C
  • B 𝑒 𝑦 + 𝑥 𝑦 + 2 𝑒 𝑦 = s i n t a n c o s C
  • C 𝑒 𝑦 + 𝑥 2 𝑦 = c o s s e c C
  • D 2 𝑒 𝑦 + 𝑥 𝑦 = s i n t a n C
  • E 𝑒 𝑦 + 𝑥 𝑦 = s i n t a n C

Q17:

Is the differential equation ( 𝑥 + 𝑦 ) 𝑥 + 𝑥 𝑦 + 𝑒 𝑦 = 0 c o s l n d d exact?

  • A no
  • B yes

Q18:

Is the differential equation 2 𝑥 𝑦 3 𝑦 𝑥 𝑥 + 2 𝑦 𝑥 𝑥 𝑦 + 1 𝑦 𝑦 = 0 d d exact?

  • Ano
  • Byes

Q19:

Solve the exact differential equation 2 𝑥 𝑦 3 𝑦 𝑥 𝑥 + 2 𝑦 𝑥 𝑥 𝑦 + 1 𝑦 𝑦 = 0 d d .

  • A 𝑥 𝑦 + 3 𝑦 5 𝑥 + 2 𝑦 = C
  • B 𝑥 𝑦 + 3 𝑦 5 𝑥 + 1 𝑦 = C
  • C 𝑥 𝑦 + 𝑦 𝑥 + 2 𝑦 = C
  • D 𝑥 𝑦 + 𝑦 𝑥 = C
  • E 𝑥 𝑦 + 𝑦 𝑥 + 𝑦 = C

Q20:

Solve the exact differential equation 3 𝑥 𝑦 + 𝑦 𝑥 + 3 𝑥 𝑦 + 𝑦 + 4 𝑥 𝑦 𝑦 = 0 d d .

  • A 𝑥 𝑦 + 𝑦 𝑥 + 𝑦 4 = C
  • B 𝑥 𝑦 + 𝑦 𝑥 = C
  • C 𝑥 𝑦 + 𝑦 𝑥 + 4 𝑦 = C
  • D 𝑥 𝑦 + 𝑦 𝑥 + 𝑦 = C
  • E 𝑥 𝑦 + 𝑥 𝑦 + 𝑦 5 = C

Q21:

Solve the exact differential equation ( 4 𝑥 𝑦 ) 𝑥 + ( 6 𝑦 𝑥 ) 𝑦 = 0 d d .

  • A 4 3 𝑥 + 2 𝑥 𝑥 𝑦 + 1 2 𝑦 = C
  • B 2 𝑥 + ( 6 𝑥 ) 𝑦 = C
  • C 2 𝑥 + ( 1 𝑥 ) 𝑦 = C
  • D 2 𝑥 𝑥 𝑦 = C
  • E 2 𝑥 𝑥 𝑦 + 3 𝑦 = C

Q22:

Solve the exact differential equation 3 𝑥 + 2 𝑦 𝑥 + 4 𝑥 𝑦 + 6 𝑦 𝑦 = 0 d d .

  • A 4 𝑥 𝑦 + 4 3 𝑦 = C
  • B 𝑥 + 2 𝑥 𝑦 + 4 𝑥 𝑦 + 4 𝑦 = C
  • C 𝑥 + 2 𝑥 𝑦 + 2 𝑦 = C
  • D 𝑥 + 2 𝑥 𝑦 = C
  • E 𝑥 + 2 𝑦 𝑥 + 6 𝑦 = C

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