Ordinary Differential Equations/Verifying Solutions to ODEs

Checking that a proposed function solves an ODE involves checking that the function satisfies the relation given by the ODE on the domain of interest, which is implicitly taken to be the whole real line in many cases. A function that solves an ODE is called a solution to the ODE. For example, consider the ODE

$$\frac{dy}{dx}=0.$$

We propose that the following function solves the differential equation: $y(x)=5$. Indeed, we find that

$$\frac{d}{dx}(5)=0,$$

so $y(x)=5$ solves this ODE. However, this is not the only function that solves this ODE. Examining the above calculation, the precise numerical value of the constant function was not used in the calculation, just the fact that the derivative of a constant function is zero. Therefore, any constant function would solve that ODE.

Now consider another example, with the ODE being

$$\frac{d^{2}y}{d{x}^2}=-4y$$

We claim that the function $y(x)=\sin (2x)$ solves this equation. We compute

$$\frac{d^{2}y}{d{x}^2}=\frac{d^2}{d{x}^2}(\sin (2x))=\frac{d}{dx}\left(\frac{d}{dx}(\sin (2x))\right)=\frac{d}{dx}(2\cos (2x))=-4\sin (2x)=-4y$$,

and note that this does satisfy the ODE, and is therefore a solution. Again, this is not the only solution to the ODE. For example, any constant multiple of $\sin (2x)$ would also solve this ODE.

Most of an ODEs course is devoted to the more difficult problem of finding an unknown function that satisfies a given ODE, and confirming that all possible solutions are known. Nonetheless, it is important to know what is meant by a function solving an ODE and how to check that a function solves an ODE before proceeding with this.