Page 2 - CH 9 Differential Equation Lession Notes

Sol :
The given differential equation is

2
(1 - x ) (i)
The given function is (ii)

. (1 mark)

my (using (ii))
Squaring both sides, we get,

2
(1- x ) (1 mark)
Differentiating again with respect to x, we get,

2
(1 - x )

is a solution of the given differential equation.
9.4 Formation of a Differential Equation whose General Solution is given
9.4.1 Differential equation that will represent a given family of curves
x
Example: Find the differential equation of the family of curves y = e (Acosx + Bsinx).
Sol:
x
We have, y = e (Acosx + Bsinx)
/ x x
y = e (Acosx + Bsinx) + e (-Asinx + Bcosx)
x
= y + e (-Asinx + Bcosx)
Differentiating with respect to x and suitable rearrangement we get the required differential equation.

NOTE:
The order of a differential equation representing a family of curves is
same as the number of arbitrary constants present in the equation corresponding to the family of curves.
9.5.1 Differential equations with variables separable
Variable separable method is used to solve such an equation in which variables can be separated completely
i.e. terms containing y should remain with dy and terms containing x should remain
with dx
Example: Solve:

Sol :
The given D.E. is

(i)