Page 193 - Foundations Of Differential Calculus

176 9. On Differential Equations
II. Let
y
y = a cos .
x
Then
y y dy −xdy + ydx y
= cos and = sin .
a x a x 2 x
Since
y y
cos = ,
x a
we have

2
y a − y 2
sin = .
x a
When we substitute this value into the differential equation, we have
2

dy (ydx − xdy) a − y 2
= ,
a ax 2
or
2 2 2
x dy =(ydx − xdy) a − y .

III. Let y = m sin x + n cos x. After the first differentiation we have dy =
mdx cos x − ndx sin x. When we keep dx constant and differentiate
2
2
2
again the result is d y = −mdx sin x − ndx cos x. When this equa-
2
2
tion is divided by the given one we have d y/y = −dx ,or
2 2
d y + ydx =0,
from which not only the sine and cosine have been eliminated, but
also the constants m and n.
IV. Let y = sin ln x. Then arcsin y =ln x, and by differentiation we have
dy dx
= .
2
1 − y x
2
2
2
2
2
When each side is squared, the result is x dy = dx −y dx . When we
2
2
let dx be constant, by another differentiation we obtain 2x dy d y +
2
2
2xdxdy = −2ydx dy,or
2 2 2
x d y + xdxdy + ydx =0.
V. Let y = ae mx sin nx. Then by differentiation,
dy = mae mx dx sin nx + nae mx dx cos nx.
When this is divided by the given equation we have