Order and Degree Examples of Differential Equations
5. FORMATION OF A DIFFERENTIAL EQUATION :
If an equation in independent and dependent variables having some arbitrary constant is given, then a differential equation is obtained as follows :
Differentiate the given equation say f (x, y, c1) = 0 w. r. t. the independent variable (say x) as many times as the number of arbitrary constants in it.
Eliminate the arbitrary constants.
The eliminant is the required differential equation.
Note : A differential equation represents a family of curves all satisfying some common properties. This can be considered as the geometrical interpretation of the differential eauation.
Examples :
(i) From the differential equation of family of lines concurent at the origin y = mx ; (dy/dx)=m
Þ y=(dy/dx).xÞ x dy –y dx=0
Note that the order is 1, same as number of constants.
(ii) Differential equation of all concentric circles the origin
X2 + y2 = r2 Þ x dx + ydy = 0
Note that the order is 1.
(iii) Differential equation of all circles touching the x axis at the origin and centre on y- axis
X2 + (y – a)2 = a2
X2 + y2 – 2ay = 0
differentiating, 2x +2y =2a (dy/dx) Þ a= {(x+y)/(dy/dx)}
(x2+ y2) (dy/dx) =2y (order is one again and degree 1)
(iv) Form the differential equations of the family of parabolas with focus at the oricin and axis of symmetry along the x-axis, equation of the parabola is
(x2 + y2) = (x + 2A)2
Y2 = 4A (A + x)
2y(dy/dx) =4A
Y(dy/dx) =2A
Hence y2 =(y (dy/dx))2 +2y(dy/dx).x or y2=y2 (dy/dx)2 +2xy(dy/dx)
(v) Form the differential equations of the lines situated at a constant distance p from the origin.
Y=mx+pÖ(1+m2)(Tangent to x2 + y2 = p2)
where m=(dy/dx)
Note that order is 1 and degree is 2.
[Alternative : Consider the line x cosa + y sina = p where p is a constant. Differentiate w. r. t. x i. e. cosa + y’ sina = 0 Þ y’ = – cota. Put in the given line
(vi) Form the differential of the system of rectangular hyperbola
x y = c2 ; diff., xdy + ydx = 0
Þ (dy/dx) =-(y/x)
Note that the radius vector and tangent at P are equally inclined to the x-axis
(vii) y2 = 2c (x+Ö c)
(viii) Form the differential equation of the family of concentric ellipse ax2 + by2 = 1 with principal axes along the co-ordinates axes.
Note that the order is 2.
(ix) Family of parabolas with axis of symmetry | | to y-axis ; y = ax2 + by + c ; diff. thrice . Þ (d3y/dx3)=0
