Worksheet: Exact Differential Equations

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

Q1:

Is the differential equation ο€Ήπ‘₯+𝑦π‘₯+ο€½π‘₯+𝑦1+3𝑦𝑦=0tandd exact?

  • Ano
  • Byes

Q2:

Solve the exact differential equation ο€Ή2π‘₯𝑦+3π‘₯π‘₯+ο€Ή2π‘₯𝑦+4𝑦𝑦=0dd.

  • Aπ‘₯𝑦+π‘₯+2π‘₯𝑦+4π‘¦βˆ’2π‘₯𝑦=C
  • Bπ‘₯𝑦+π‘₯=C
  • Cπ‘₯𝑦+π‘₯+4𝑦=C
  • D2π‘₯𝑦+π‘₯+π‘¦βˆ’23π‘₯π‘¦βˆ’3π‘₯𝑦=οŠͺC
  • Eπ‘₯𝑦+π‘₯+𝑦=οŠͺC

Q3:

Solve the exact differential equation (1+𝑦𝑒)π‘₯+(2𝑦+π‘₯𝑒)𝑦=0ο—ο˜ο—ο˜dd.

  • Aπ‘₯+𝑒+2𝑦=ο—ο˜C
  • Bπ‘₯+π‘¦βˆ’π‘¦+ο€Ό2+1π‘₯οˆπ‘’=οŠ¨ο—ο˜C
  • Cπ‘₯+2𝑦+(2+π‘₯)𝑒=ο—ο˜C
  • Dπ‘₯+𝑒=ο—ο˜C
  • Eπ‘₯+𝑒+𝑦=ο—ο˜οŠ¨C

Q4:

Is the differential equation 2π‘₯βˆ’3𝑦2π‘₯𝑦π‘₯+3π‘¦βˆ’2π‘₯3π‘₯𝑦𝑦=0dd exact?

  • Ano
  • Byes

Q5:

Is the differential equation ο€Όπ‘₯+𝑦π‘₯π‘₯+𝑦+5π‘₯𝑦=0dlnd exact?

  • Ayes
  • Bno

Q6:

Solve the exact differential equation (π‘₯+𝑦)π‘₯+ο€½π‘₯𝑦+𝑒𝑦=0coslndd.

  • Aβˆ’π‘₯+π‘₯𝑦+𝑒=sinlnC
  • BsinlnCπ‘₯+π‘₯𝑦+𝑒=
  • CsinlnCπ‘₯+π‘₯𝑦=
  • Dβˆ’π‘₯βˆ’π‘₯𝑦+𝑒=sinlnC
  • Eβˆ’π‘₯+π‘₯𝑦=sinlnC

Q7:

Solve the exact differential equation 2π‘₯βˆ’3𝑦2π‘₯𝑦π‘₯+3π‘¦βˆ’2π‘₯3π‘₯𝑦𝑦=0dd.

  • Aπ‘₯𝑦+72𝑦π‘₯=C
  • Bπ‘₯𝑦+𝑦π‘₯=C
  • C𝑦+𝑦π‘₯=C
  • Dβˆ’23𝑦+π‘₯=C
  • Eβˆ’23π‘₯𝑦+π‘₯=C

Q8:

Solve the exact differential equation ο€»π‘₯+𝑦π‘₯π‘₯+𝑦+π‘₯𝑦=0dlnd.

  • Aπ‘₯4+𝑦π‘₯+𝑦=οŠͺlnC
  • Bπ‘₯4+2𝑦π‘₯+𝑦3βˆ’π‘₯π‘¦βˆ’π‘¦2π‘₯=οŠͺlnC
  • Cπ‘₯4+𝑦π‘₯+π‘¦βˆ’π‘₯=οŠͺlnC
  • Dπ‘₯4+𝑦π‘₯+𝑦3=οŠͺlnC
  • Eπ‘₯4+𝑦π‘₯=οŠͺlnC

Q9:

Solve the exact differential equation ο€Ήπ‘₯+𝑦π‘₯+ο€½π‘₯+𝑦1+𝑦𝑦=0tandd.

  • Aπ‘₯2+π‘₯𝑦=tanC
  • Bπ‘₯2+π‘₯𝑦+ο€Ή1+𝑦=tanlnC
  • Cπ‘₯2+π‘₯𝑦+ο€Ή1+𝑦2=tanlnC
  • Dπ‘₯2+π‘₯𝑦+𝑦1+𝑦=tanC
  • E2π‘₯+π‘₯𝑦+𝑦1+𝑦=tanC

Q10:

Is the differential equation ο€Ή3π‘₯+2𝑦π‘₯+ο€Ή4π‘₯𝑦+6𝑦𝑦=0dd exact?

  • AYes
  • BNo

Q11:

Find the general solution for the following exact differential equation: 1π‘¦βˆ’ο€½π‘₯𝑦𝑦π‘₯=0.dd

  • A𝑦=𝑐π‘₯
  • B𝑦=π‘βˆšπ‘₯
  • C𝑦=π‘₯𝑐
  • D𝑦=π‘₯+𝑐

Q12:

Solve the exact differential equation ο€Ή6π‘₯π‘¦βˆ’π‘¦ο…π‘₯+ο€Ή4𝑦+3π‘₯βˆ’3π‘₯𝑦𝑦=0dd.

  • A4π‘₯𝑦+π‘₯βˆ’32π‘₯𝑦+4𝑦=C
  • B4π‘₯𝑦+π‘₯βˆ’32π‘₯𝑦+2𝑦=C
  • C3π‘₯π‘¦βˆ’π‘₯𝑦+2𝑦=C
  • D3π‘₯π‘¦βˆ’π‘₯𝑦=C
  • E3π‘₯π‘¦βˆ’π‘₯𝑦+4𝑦=C

Q13:

Solve the exact differential equation (2π‘₯+3𝑦)π‘₯+(3π‘₯+2𝑦)𝑦=0dd.

  • Aπ‘₯+3π‘₯𝑦=C
  • Bπ‘₯+3π‘₯𝑦+𝑦=C
  • Cπ‘₯+3π‘₯𝑦+2𝑦=C
  • Dπ‘₯+4π‘₯π‘¦βˆ’12𝑦=C
  • Eπ‘₯+3π‘₯𝑦+π‘₯βˆ’π‘¦=C

Q14:

Is the differential equation (2π‘₯+3𝑦)π‘₯+(3π‘₯+2𝑦)𝑦=0dd exact?

  • Ano
  • Byes

Q15:

Solve the exact differential equation 𝑦π‘₯+3π‘₯𝑦𝑦=0dd.

  • Aπ‘₯𝑦=C
  • Bπ‘₯𝑦+32π‘₯𝑦=C
  • C2π‘₯π‘¦βˆ’π‘¦4=οŠͺC
  • D32π‘₯𝑦=C
  • E𝑦=C

Q16:

Solve the exact differential equation (𝑒𝑦+𝑦)π‘₯+𝑒𝑦+π‘₯𝑦𝑦=0ο—ο—οŠ¨sintandcossecd.

  • A𝑒𝑦+π‘₯2𝑦=ο—οŠ¨οŠ¨cossecC
  • B𝑒𝑦+π‘₯2π‘¦βˆ’2𝑒𝑦=ο—οŠ¨οŠ¨ο—cossecsinC
  • C𝑒𝑦+π‘₯𝑦=sintanC
  • D𝑒𝑦+π‘₯𝑦+2𝑒𝑦=sintancosC
  • E2𝑒𝑦+π‘₯𝑦=sintanC

Q17:

Is the differential equation (π‘₯+𝑦)π‘₯+ο€½π‘₯𝑦+𝑒𝑦=0coslndd exact?

  • Ayes
  • Bno

Q18:

Is the differential equation ο€Ύ2π‘₯π‘¦βˆ’3𝑦π‘₯π‘₯+ο€Ώ2𝑦π‘₯βˆ’π‘₯𝑦+1βˆšπ‘¦ο‹π‘¦=0οŠͺdd exact?

  • Ayes
  • Bno

Q19:

Solve the exact differential equation ο€Ύ2π‘₯π‘¦βˆ’3𝑦π‘₯π‘₯+ο€Ώ2𝑦π‘₯βˆ’π‘₯𝑦+1βˆšπ‘¦ο‹π‘¦=0οŠͺdd.

  • Aπ‘₯𝑦+𝑦π‘₯+βˆšπ‘¦=C
  • Bπ‘₯𝑦+𝑦π‘₯+2βˆšπ‘¦=C
  • Cπ‘₯𝑦+3𝑦5π‘₯+2βˆšπ‘¦=C
  • Dπ‘₯𝑦+3𝑦5π‘₯+1βˆšπ‘¦=C
  • Eπ‘₯𝑦+𝑦π‘₯=C

Q20:

Solve the exact differential equation ο€Ή3π‘₯𝑦+𝑦π‘₯+ο€Ή3π‘₯𝑦+𝑦+4π‘₯𝑦𝑦=0οŠͺοŠͺdd.

  • Aπ‘₯𝑦+𝑦π‘₯=C
  • Bπ‘₯𝑦+π‘₯𝑦+𝑦5=οŠͺC
  • Cπ‘₯𝑦+𝑦π‘₯+4𝑦=οŠͺC
  • Dπ‘₯𝑦+𝑦π‘₯+𝑦=οŠͺοŠͺC
  • Eπ‘₯𝑦+𝑦π‘₯+𝑦4=C

Q21:

Solve the exact differential equation (4π‘₯βˆ’π‘¦)π‘₯+(6π‘¦βˆ’π‘₯)𝑦=0dd.

  • A2π‘₯+(6βˆ’π‘₯)𝑦=C
  • Bβˆ’43π‘₯+2π‘₯βˆ’π‘₯𝑦+12𝑦=C
  • C2π‘₯βˆ’π‘₯𝑦+3𝑦=C
  • Dβˆ’2π‘₯+(1βˆ’π‘₯)𝑦=C
  • E2π‘₯βˆ’π‘₯𝑦=C

Q22:

Solve the exact differential equation ο€Ή3π‘₯+2𝑦π‘₯+ο€Ή4π‘₯𝑦+6𝑦𝑦=0dd.

  • A4π‘₯𝑦+43𝑦=C
  • Bπ‘₯+2𝑦π‘₯+6𝑦=C
  • Cπ‘₯+2π‘₯𝑦+2𝑦=C
  • Dπ‘₯+2π‘₯𝑦+4π‘₯𝑦+4𝑦=C
  • Eπ‘₯+2π‘₯𝑦=C

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