Formation of differential equations (Contd)

Misc > DIFFERENTIAL EQUATIONS > Formation of differential equations > Formation of differential equations (Contd)

Example 5

Form the differential equation corresponding toy² ~ m (a² ~x²) by eliminating m and a. .. --

Sol. Differentiating the given diff. equation w.r.t. we get

2y^- m(-2x) = -2 mx or ..it)

-dx dx :

d²y dy dy d²y f dy²

Differentiating again, we get y ^or > ⁺ (^J. ⁼ " ^m

Substituting the value of m in (/), we get

d²v (dv² v dv d²y (dy² y dy ^y lb? ⁺1 dx) ⁼x'dx °^{r y 2 +}1 dx J ~ dx ^{= is re<}l^u*^red diff. equation-

Example 6

Find the differential equation of all the circ{les} in the first quadrant which touch the coordinates axes. (H.B.)

Sol. The equation of circ{les} in the first quadrant which touch the coordinate axes is (x - a)² + (y - a)² -a² , .. .(0

where a is an arbitrary constant. Since the equation contains one arbitrary constant, we shall differentiate it only once and get a differential equation of first order.

Differentiating (/) w.r.t. x, we get,

^S .

Substituting the value of a in (0, we get

(x-y)²k² + x~y)² = (x + ky)² => (x-y)² (k² + I) = (x + ky)² _;=> (x-yi^{1 = +} . This is the required differential equation.'

Example 7

Form the differential equation of all the circ{les} which pass through the origin and whose centre lies on .y-axis. (A.P.)

Sol. The general equation of the circ{les} with the given properties is

(x-0)² + (y-a)² = a , or x² + (y- a)² = a² ...(/)

or x² +y² ~ 2 ay = a² -a² = 0 ...(«")

We get different members of the family for different values of a. , Differentiating (/) w.r.t. x, we get

2x+2(y~a)y' = 0 (ut

x + yy'

or x = - (y-a)y'=-yy'+ ay' a=-~~

r y

2 2 f x + yy j Putting this value of a in (ii), we get x + y -2 \—~ ^

or x² + y - 2 xy ~ 2y² = 0 (substitutingy' = or x-y² 2 xy ~~ = 0, which is

dy dy

required differential equation.

Example 8

Assume that a spherical rain drop evaporates at a rate proportional to its surface

area. Form a differential equation involving the rate of change of the radius of the rain drop.

(LLT. 1997)

Formation of differential equations (Contd)

Sol. Let r denote the radius (in mm) of the rain drop after t minutes., Since the radius is decreasing as t increases, the rate of change of r must be negative.

If V denotes the volume of the rain drop and S its surface area, we have

V= and S=4nr¹. It is also given that ^ « S. Hence j 3r^z ~ = _ k • 4nr²

dr . .

=> —j^ = - k. This is the required differential equation. _s

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