5· y=ex (a cos x + b sin x)
The given function is,
y = e x ( a cos x + b sin x ) (1)
Differentiate both side with respect to x ,
d ( y ) d x = d d x e x ( a cos x + b sin x )
Use product rule and differentiate,
y ′ = ( a cos x + b sin x ) d d x e x + e x d d x ( a cos x + b sin x ) y ′ = e x ( a cos x + b sin x ) + e x ( a d d x cos x + b d d x sin x ) y ′ = e x ( a cos x + b sin x ) + e x ( − a sin x + b cos x ) y ′ = e x [ ( a + b ) cos x − ( a − b ) sin x ] (2)
Again differentiate,
d d x y ′ = d d x e x [ ( a + b ) cos x − ( a − b ) sin x ]
Again use product rule,
y ″ = e x [ ( a + b ) cos x − ( a − b ) sin x ] + e x [ − ( a + b sin x − ( a − b ) cos x ) ] = e x [ 2 b cos x − 2 a sin x ] = 2 e x [ b cos x − a sin x ] y ″ 2 = e x ( b cos x − a sin x ) (3)
Add equation (1) and (3),
y + y ″ 2 = e x ( a cos x + b sin x ) + e x ( b cos x − a sin x ) = e x [ a cos x + b sin x + b cos x − a sin x ] = e x [ cos x ( a + b ) − sin x ( a − b ) ]
From equation we can write it as:
y + y ″ 2 = y ′ 2 y + y ″ = 2 y ′ y ″ − 2 y ′ + 2 y = 0
Thus, the required differential equation of the curve is y ″ − 2 y ′ + 2 y = 0 .
The given function is,
Differentiate both side with respect to
Use product rule and differentiate,
Again differentiate,
Again use product rule,
Add equation (1) and (3),
From equation we can write it as:
Thus, the required differential equation of the curve is
