Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b. y = ex (a cos x + b sin x)

Explanation

Given , `y=e^(x)(acosx+bsinx)`……….`(1)`
divide both sides by `e^(x)`,
`e^(-x)y=(a cos x+b sin x)`……..`(2)`
differentiate w.r.t.x,
`e^(-x)y'+ye^(-x)(-1)=-a sin x+b cos x`
Again, differentiate w.r.t.x,
`e^(-x)(d)/(dx)(y')+y'(d)/(dx)(e^(-x))-[y(d)/(dx)e^(-x)+e^(-x)(d)/(dx)y]`
`=-a cosx-b sinx`
`implies e^(-x)y'+y'e^(-x)(-1)-[ye^(-x)(-1)+e^(-x)y']`
`=-a cos x-b sin x`
`implies e^(-x)y''-2y'e^(-x)+ye^(-x)=-(ye^(-x))` [from equation `(2)`]
`implies e^(-x)[y''-2y'+2y]=0`
`implies y''-2y'+2y=0` (divide by `e^(-x)`)
which is the required differential equation.