Questions on Cauchy-Euler Equations with Answers

Practice Questions on Cauchy-Euler Equations with Answers

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Questions on Cauchy-Euler Equations with Answers


Q1. Solve the differential Equation :

x3 (d3y/dx3) + 3x2 (d2y/dx2) + x (dy/dx) + y = log x + x

  • (a) y = c1 x-1 + x1/2 [c2 cos {(√3/2) logx} + c3 sin{(√3/2) logx}] + x/2 + logx
  • (b) y = c1 x + x1/2 [c2 sin {(√3/2) logx} + c3 cos {(√3/2) logx}] + x/2 + logx
  • (c) y = c1 x-1 + x1/2 [c2 cos {(√3/2) logx} + c3 cos{(√3/2) logx}] + x/2 + logx
  • (d) y = c1 x + x1/2 [c2 sin {(√3/2) logx} + c3 sin{(√3/2) logx}] + x/2 + logx

Q2. Solve the differential Equation :

3x2y2 – 5xy1 + 5y = sin (log x).

  • (a) y = c1x + c2 x7/3 + (1/8) × [cos (log x) + cos (log x)].
  • (b) y = c1x + c2 x5/3 + (1/16) × [sin (log x) + cos (log x)].
  • (c) y = c1x + c2 x5/3 + (1/32) × [sin (log x) + sin (log x)].
  • (d) y = c1x + c2 x7/3 + (1/2) × [sin (log x) + cos (log x)].

Q3. Solve the differential Equation :

{x2D2 – (2m – 1) xD + (m2 + n2)} y = n2 xm log x

  • (a) y = xm [c1 sin (n log x) + c2 sin (n log x)] + xm log x
  • (b) y = xm [c1 cos (n log x) – c2 sin (n log x)] + mx log x
  • (c) y = xm [c1 cos (n log x) + c2 sin (n log x)] + xm log x
  • (d) y = xm [c1 sin (n log x) – c2 sin (n log x)] + mx log x

Q4. Solve the differential Equation :

(x2D2 – xD + 4) y = cos (log x) + x sin (log x).

  • (a) y = x [ c1 cos(√3 log x) – c2 sin(√3 log x)] + (1/13) [3cos(log x) + 2sin(log x)] + (x/2) 2sin(logx)
  • (b) y = x [ c1 cos(√3 log x) + c2 sin(√3 log x)] + (1/13) [3cos(log x) – 2sin(log x)] + (x/2) 2sin(logx)
  • (c) y = x [ c1 cos(√3 log x) – c2 sin(√3 log x)] + (1/13) [3cos(log x) + 2sin(log x)] + (x/2) 2sin(logx)
  • (d) y = x [ c1 cos(√3 log x) + c2 sin(√3 log x)] + (1/13) [3cos(log x) – 2sin(log x)] + (x/2) 2sin(logx)

Q5. Solve the differential Equation :

x2 (d2y/dx2) – 2x (dy/dx) + 2y = x + x2 log x + x3

  • (a) y = c1x + c2x2 – x log x + x3/2 + (x2/2) 2 [(log x)2 – 2log x].
  • (b) y = c1x + c2x2 + x log x + x/2 – (x2/2) 2 [(log x)2 – 2log x].
  • (c) y = c1x + c2x2 – x log x + x3/2 + (x2/2) 2 [(log x)2 + 2log x].
  • (d) y = c1x + c2x2 + x log x + x/2 – (x2/2) 2 [(log x)2 + 2log x].

Q6. Solve the differential Equation :

(x4D3 + 2x3D2 – x2D + x) y = 1.

  • (a) y = (c1 + c2 log x) x + c3 x + (1/4) 2 x–1 log x
  • (b) y = (c1 + c2 log x) x + c3 x–1 + (1/4) 2 x–1 log x
  • (c) y = (c1 – c2 log x) x + c3 x–1 + (1/4) 2 x log x
  • (d) y = (c1 + c2 log x) x – c3 x + (1/4) 2 x log x

Q7. Solve the differential Equation :

x2D2y – 3x Dy + 5y = x2 sin log x

  • (a) y = x2 (c1 cos log x + c2 sin log x) – (x/2) × log x cos(log x)
  • (b) y = x2 (c1 cos log x – c2 sin log x) + (x2/2) × log x cos(log x)
  • (c) y = x2 (c1 cos log x + c2 sin log x) – (x2/2) × log x cos(log x)
  • (d)y = x2 (c1 cos log x – c2 sin log x) + (x/2) × log x cos(log x)

Q8. Solve the differential Equation :

x3 (d3y/dx3) + 2x2 (d2y/dx2) + 2y = 10(x + 1/x)

  • (a) y = c1x–1 + x (c2 cos log x – c3 sin log x) + 5x – 2x–1 log x
  • (b) y = c1x – x (c2 cos log x – c3 sin log x) – 5x + 2x–1 log x
  • (c) y = c1x–1 + x (c2 cos log x – c3 sin log x) + 5x – 2x–1 log x
  • (d) y = c1x–1 + x (c2 cos log x + c3 sin log x) + 5x + 2x–1 log x

Q9. Solve the differential Equation :

(x4D4 + 6x3D3 + 9x2D2 + 3xD + 1) y = (1 + log x)2

  • (a) y = (c1 + c2log x) cos log x + (c3 + c4 log x)sin log x + (log x)2 + 2 log x – 3.
  • (b) y = (c1 – c2log x) cos log x + (c3 + c4 log x)cos log x – (log x)2 + 2 log x – 3.
  • (c) y = (c1 – c2log x) cos log x + (c3 + c3 log x)sin log x – (log x)2 + 2 log x – 3.
  • (d) y = (c1 + c2log x) cos log x + (c3 + c4 log x)sin log x + (log x) – 2 log x – 3.

Q10. Solve the differential Equation :

(x2D2 + xD + 1) y = log x. sin log x

  • (a) y = c1cos log x – c2sin log x + (1/4) log x sin(log x) + (1/4) (log x)2cos (log x)
  • (b) y = c1cos log x + c2sin log x + (1/4) log x sin(log x) – (1/4) (log x)2cos (log x)
  • (c) y = c1cos log x – c2sin log x + (1/4) log x sin(log x) – (1/4) (log x)2 cosx (log x)
  • (d) y = c1cos log x + c1sin log x + (1/4) log x sinx (log x) – (1/4) (log x)2cos (log x)

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Answer Key

01.(a)06.(b)
02.(b)07.(c)
03.(c)08.(d)
04.(d)09.(a)
05.(a)10.(b)

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