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
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
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
Problem 2 :
y' = x/(y2 + 1)
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
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
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
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
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
Problem 8 :
The solution of the differential equation
2x (dy/dx) - y = 3
represents
a) straight lines b) circles c) parabola d) ellipse
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?
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?
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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