Question

In Problems15 and 16 interpret each statement as a differential equation.

On the graph of $y=\varphi \left(x\right)$the slope of the tangent line at a point $P(x,y)$ is the square of the distance from P(x,y) to the origin.

Short answer

The differential equation is $\frac{dy}{dx}=x^2+y^2$.

Step-by-step solution

Define a derivative of the function

The sensitivity of the function is derived by the derivative of a function of a real variable for modifying the argument.

Calculus uses derivatives as a fundamental tool. When a derivative of a single-variable function exists at a given input value, it is the slope of the tangent line to the function's graph at that point.

Determine the differential equation for the height

Let the slope of the tangent line of the graph of $y=\varphi \left(x\right)$at the point P(x,y) is $\frac{dy}{dx}$.

Using the Pythagoras theorem, the square of the distance of P(x,y) to the origin is $x^2+y^2$.

Hence, the statement describes the differential equation,

$\frac{dy}{dx}=x^2+y^2$.