Given an initial function $y(x)=C_1x+C_2\sin(x)$ how can I derive a second order differential equation with no arbitrary constants?
I have attempted by finding the derivatives: $y'=C_1+C_2\cos(x)$; $y''=-C_2\sin(x)$; and then forming an ODE of the form $y''+Ay'+By=0,$ but I am unable to find values for $A$ and $B$ which satisfy this.
Best Answer
To eliminate the first constant, write
$$\frac yx=C_1+C_2\frac{\sin x}x$$
then taking the derivative,
$$\frac{y'x-y}{x^2}=C_2\frac{\cos x\, x-\sin x}{x^2}.$$
To eliminate the second constant, now write
$$\frac{y'x-y}{\cos x\,x-\sin x}=C_2.$$
After differentiation and simplification, the numerator yields
$$y''(\cos x\,x-\sin x)+(y'x-y)\sin x=0.$$