18. Find the solution $y(x)$ of the differential equation $y^{\prime}=$ $=\cos x$, satisfying the condition $y(0)=1$.
Solution. The general solution of this equation is found by integration: $y=\sin x+C$. By substituting the initial conditions and determining $C=1$, we find the particular solution (solution of the Cauchy problem): $y=\sin x+1$.

The problems considered above on finding the equation of a curve for which the slope of the tangent is equal to $x$, on finding the antiderivative of a given function, lead to simple differential equations. Similar differential equations also arise from problems related to the rate of change of a function, since the rate is the derivative of the given function with respect to time. For example, if it is necessary to find the law of motion $s(t)$ given the velocity, then it is necessary to solve a differential equation of the form $\frac{d s}{d t}=f(t)$, where $\frac{d s}{d t}$ is the velocity of motion.

Let's consider some specific problems leading to differential equations as examples.