The dif is a:
We know to calculate degree or order, the powers of derivatives should be integer.
on squaring both sides, we get a differential equation with integers as the power of derivatives
so order is the highest derivative, which is 2
and degree is the power of highest degree which is again 2
The differential equation for the equation is :
y = Acos(αx) + Bsin(αx)
dy/dx = -Aαsin(αx) + Bαcos(αx)
d2y/dx2 = -Aα2cos(αx) - Bα2sin(αx)
= -α2(Acos(αx) + Bsin(αx))
= -α2 * y
d2y/dx2 + α2*y = 0
Formation of the differential equation of the family of curves represented by y = Ae2x + Be-2x is :
y = Ae2x + Be-2x
dy/dx = 2Ae2x – 2Be-2x
d2y/dx2 = 4Ae2x + 4Be-2x
d2y/dx2 – 4y = 0
The degree of the differential equation
Given equation is :
(dy/dx)2 + 1/(dy/dx) = 1
((dy/dx)3 + 1)/(dy/dx) = 1
(dy/dx)3 +1 = dy/dx
So, final equation is
(dy/dx)3 - dy/dx + 1 = 0
So, degree = 3
Differential equation representing the family of curves given by y = ax + x2 is:
The answer is C. We eliminate constants.
Differentiating with respect to x,
The order of the differential equation:
Order of the D.E. is 3
Order of a differential equation is the order of the highest derivative present in the equation.
Formation of the differential equation corresponding to the ellipse major axis 2a and minor axis 2b is:
Equation of ellipse :
x2/a2 + y2/b2 = 1
Differentiation by x,
2x/a2 + (dy/dx)*(2y/b2) = 0
dy/dx = -(b2/a2)(x/y)
-(b2/a^2) = (dy/dx)*(y/x) ----- eqn 1
Again differentiating by x,
d2y/dx2 = -(b2/a2)*((y-x(dy/dx))/y2)
Substituting value of -b2/a2 from eqn 1
d2y/dx2 = (dy/dx)*(y/x)*((y-x(dy/dx))/y2)
d2y/dx2 = (dy/dx)*((y-x*(dy/dx))/xy)
(xy)*(d2y/dx2) = y*(dy/dx) - x*(dy/dx)2
(xy)*(d2y/dx2) + x*(dy/dx)2- y*(dy/dx) = 0
The differential equation is a solution of the equation:
Solving second order differential equation with variable coefficients becomes a bit lengthy and complicated. So, its better to check by options.
On checking option A :
y = A/x + B
dy/dx = -A/x2
d2y/dx2 = (2A)/x3
d2y/dx2 + (2/x)*(dy/dx) = 0
(2A)/x3 + (2/x)*((-A)/x2) = 0
(2A - 2A)/x3 = 0
0 = 0
LHS = RHS
The differential equation is a:
3*(d2y/dx2) = [1+(dy/dx)2]3/2
On squaring both side,
9*(d2y/dx2)2 = [1+(dy/dx)2]3
The order of the equation is 2. The power of the term determining the order determines the degree.
So, the degree is also 2.
The order and degree of the differential equation: (y”)2 + (y”)3 + (y’)4 + y5 = 0 is:
The highest order derivative here is y’’. Therefore the order of the differential equation=2.
The highest power of the highest order derivative here is 3. Therefore the order of the differential equation=3.