For the sine function we have a simple relationship between the function and its second derivative; it is equal to the negative sine function. The same relationship applies to the cosine function.

$$\begin{array}{c}y=sin\left(x\right)\to {y}^{\prime}=\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}cos\left(x\right)\to {y}^{\u2033}=-sin\left(x\right)\Rightarrow {y}^{\u2033}=-y\hfill \\ y=cos\left(x\right)\to {y}^{\prime}=-sin\left(x\right)\to {y}^{\u2033}=-cos\left(x\right)\Rightarrow {y}^{\u2033}=-y\hfill \\ \hfill \end{array}$$

For the trigonometric functions the differential equation expresses the fact, that the absolute value of the derivative is equal to the function value having opposite sign. What does this mean in concrete terms?

If the function value $y$ is positive and large, the curvature is negative and large, leading quickly to smaller values of $y$. If the the function value is negative and has a large absolute value, the large positive curvature quickly leads to larger value. If the function value is small, the curvature is also small and therefore an increase or decrease continues nearly linearly as at a turning point.

The negative relationship between the function and their curvature thus leads to oscillating behaviour. You are encouraged to confirm these statements in the last two figures.

The fact, that both trigonometric functions $sinx$ and $cosx$ satisfy the same differential equations shows their close relationship as oscillating functions. It follows immediately, that the sum of sine and cosine functions satisfies the same differential equation. (Also check that this sum is , according to the addition rules for trigonometric functions, identical to a phase-shifted function.) As second example we consider the exponential function for both positive and negative exponents:

$$\begin{array}{c}y={e}^{x}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\to {y}^{\prime}={e}^{x};\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}{y}^{\u2033}={e}^{x}\to \phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}{y}^{\prime}=y\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\text{and}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}{y}^{\u2033}=\phantom{\rule{0em}{0ex}}y\hfill \\ y={e}^{-x}\to {y}^{\prime}=-{e}^{-x};{y}^{\u2033}={e}^{x}\to {y}^{\prime}=-y\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\text{and}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}{y}^{\u2033}=y\hfill \\ \hfill \end{array}$$

Now the second derivative also has the same sign as the original function. What do these relationships mean in concrete terms?

The curvature is equal to the function value. The larger the function value, the larger the curvature. Any curvature already present increases with increasing $y$. If the slope (first derivative) has the same sign as the function, the function will grow faster and faster beyond any boundaries – it diverges. If the slope has the opposite sign of the function, the function decreases faster and faster to zero, it converges to $0$. The differential equation ${y}^{\u2033}=y$ describes both behaviours.

As shown for the trigonometric functions, the differential equation then is also valid for the sum of two exponential functions. If one takes exponents with different signs for the two functions , the hyperbolic functions are covered:

$$\phantom{\rule{0em}{0ex}}\text{sinh}\left(x\right)=\frac{{e}^{x}-{e}^{-x}}{2};\phantom{\rule{0em}{0ex}}\text{cosh}\left(x\right)=\frac{{e}^{x}+{e}^{-x}}{2}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}$$

Thus the differential equation ${y}^{\u2033}=y$ describes the exponential and hyperbolic functions and this common property shows their close relationship.

Differential equations describe the local, internal structures of function , their character and they are the “generators” of cohorts of related functions.