CREATING DIFFERENTIAL EQUATION WORKSHEET

Form a differential equations by eliminating arbitrary constants given in brackets against each.

(i)  y2  =  4ax       {a}          Solution

(ii)  y  =  ax2+bx+c       {a, b}            Solution

(iii) x y  =  c2       {c}          Solution

(iv) (x2/a2) + (y2/b2 )  =  1     {a , b}           Solution

(v) y  =  A e2x + Be-5x   {A , B}         Solution

(vi) y  =  (A + Bx) e3x        {A , B}    Solution

(vii) y  =  e3x {C cos 2 x + D sin 2 x}    {C , D}       Solution

(viii) y  =  emx         {m}      Solution

(ix) y = Ae2x cos (3 x + B)      {A, B}           Solution

Answer Key

1)  Therefore the required equation is y  =  2xy'.

2)  x2y'' - 2xy' + 2y - 2c  =  0

3)  xy' + y  =  0

4)  xy'' + xyy' - yy'  =  0

5)  y'' + 3y' - 10y  =  0

6)  y'' - 6y' - 5y  =  0

7)  y' - 3y  =  0

8)  y' + 7y  =  0

9)  y'' + 2y' + 2y = 0

Find the order and degree of the following differential equations.

(1)  (dy/dx) + y  =  x2         Solution

(2)  y' + y2  =  x            Solution

(3)  y'' + 3 (y')2 + y3               Solution

(4)  d2y/dx2 + x  =  √[y + (dy/dx)]             Solution

(5)  d2y/dx2 - y + (dy/dx + d3y/dx3)(3/2)  =  0       Solution

(6)  y''  =  (y - (y')3)(2/3)                   Solution

(7)  y' + (y'')2  =  (x + y'')2                  Solution

(8)  (dy/dx)2 + x = (dx/dy) + x2                   Solution

Answer Key

1)  Order  =  1,  Degree  =  1.

2)  order  =  2 and degree  =  1

3)  order  =  2 and degree  =  1

4) order  =  2 and degree  =  2

5)  order  =  3 and degree  =  3

6)  order is 2 and degree is 3.

7)  the order is 2 and degree is 1.

8)   the order is 1 and degree is 3.

Find the the following for the given differential equations.

(i) Order 

(ii) Degree

(iii) General solution

Problem 1 :

y' = 1 + x2 + y + x2y

Solution

Problem 2 :

y' = x/(y+ 1)

Solution

Problem 3 :

The order and degree of the differential equation

d2y/dx2 + (dy/dx)1/3 + x1/4 = 0

a) 2, 3    b) 3, 3    c) 2, 6     d)  2, 4

Solution

Problem 4 :

The differential equation representing the family of curves y = A cos (x + B), where A and B are parameters, is

a)  d2y/dx2 - y = 0     b)  d2y/dx2 + y = 0

c)  d2y/dx2 = 0     d)  d2y/dx2  = 0

Solution

Problem 5 :

The order and degree of the differential equation 

√sin x (dx + dy) = √cos x (dx - dy) is

a)  1, 2      b)  2, 2     c)  1, 1     d)  2, 1

Solution

Problem 6 :

The order of the differential equation of all circles with center at (h, k) and radius a is 

a)  2    b)  3     c)  4    d)  1

Solution

Problem 7 :

The differential equation of the family of curves 

y = Aex + Be-x 

where A and B are arbitrary constants is 

a)  d2y/dx2 + y = 0     b)  d2y/dx2 - y = 0

c)  dy/dx + y = 0     d)  dy/dx - y = 0

Solution

Problem 8 :

The solution of the differential equation 

2x (dy/dx) - y = 3

represents

a) straight lines   b) circles     c) parabola   d) ellipse

Solution

Answer Key

1)  y = Cex + (x^3/3) - 1

2)  y3/3 + y = x2/2 + C

3)  Order = 2 and degree = 4

4)  d2y/dx2 + y = 0

5)   1, 1 

6)  2

7)  d2y/dx2 - y = 0

8)  

(y + 3)2 = x 

So, the general solution must be a parabola.

Problem 1 :

Radioactive radium has a half-life of approximately 1599 years.

A. If 50 milligrams of pure radium is present initially, when will the amount remaining be 20 milligrams? Give your answer to the nearest year.

B. What percent of a given amount remains after 100 years?

Solution

Problem 2 :

Water is leaking out of a large barrel such that the rate of the change in the water level is proportional to the square root of the depth of the water at that time. If the water level starts at 36 inches and drops to 34 inches in 1 hour, how long will it take for all of the water to drain out of the barrel?

Solution

Answer key

1)  95.7 percent of the given amount remains after 100 years.

2)  It will take about 35.5 hours for for all of the water to drain out of the barrel.

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