# Is this sine equation solvable

#### isatis55

Hi,

I need to either solve or prove that the resolution of the following ODE is impossible;

$$\displaystyle y''(x)=a*y(x)+b*y'(x)+c*sin(d*x)*y(x)$$

Mathemtatica refuse solving it rather giving

" InverseFunction::ifun: Inverse functions are being used. Values may be lost for multivalued inverses."

Any help would be highly apreciated, thank you

Isatis55

Last edited:

#### chiro

MHF Helper
Hey isatis55.

I haven't taken DE's for a while but the output suggests that f(x) is not a unique function (with the multi-valued comment) and this implies that y(x) doesn't exist since it has to be unique by definition to give a function (functions must be unique and give one output for every input).

#### TheEmptySet

MHF Hall of Honor
Hi,

I need to either solve or prove that the resolution of the following ODE is impossible;

$$\displaystyle y''(x)=a*y(x)+b*y'(x)+c*sin(d*x)*y(x)$$

Mathemtatica refuse solving it rather giving

" InverseFunction::ifun: Inverse functions are being used. Values may be lost for multivalued inverses."

Any help would be highly apreciated, thank you

Isatis55
If you multiply the equation by $$\displaystyle -e^{-bx}$$

This gives

$$\displaystyle -e^{-bx} y''+be^{-bx}y'+ae^{-bx}y= -ce^{-bx}\sin(dx) y$$

Now the differential equation can be put in its self-adjoint form

$$\displaystyle -\frac{d}{dx}\left[ e^{-bx}\frac{dy}{dx}\right]+ae^{-bx}y= -ce^{-bx}\sin(dx) y$$

This is a Sturm-Liouville differential equation.

The boundary or initial conditions determine if the equation has solutions.

For example if you require $$\displaystyle y(0)=y'(0)=0$$

The equation will have only the trivial solution $$\displaystyle y=0$$

#### isatis55

Great, thanks a lot.

I am now altering the initial problem and I am again faced against a non linear differential equation given by

$$\displaystyle y''(x)+a*y'(x)^2+b*y'(x)+c*y(x) + c*sin(d*t) + e = 0 [\math] Do you think it is solvable? PS: I definitely need a course on ODE.$$