Plot the isoclines of the differential equation y′ = x² + y²

by RoRi

January 31, 2016

If $y$ is the solution of a differential equation, the points at which $y'$ has a constant value $C$ lie on a line for each $C$ . This line is called an isocline.

For the differential equation

$y' = x^2 + y^2$

plot the isoclines corresponding to the constant slopes $\frac{1}{2}, \ 1, \ \frac{3}{2}$ , and 2. Using these isoclines, construct a direction field for the equation and determine the shape of the integral curve which passes through the origin.

The isoclines of the differential equation $y' = x^2 + y^2$ are concentric circles centered at the origin with slope equal to the radius of the circle. The shape of the integral curve passing through the origin is cubic.

Stumbling Robot

Plot the isoclines of the differential equation y′ = x² + y²

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