My question is this;
The relation f(x,y)=c where c is constant, represents the family of level curves of f in the xy-plane, what is the geometrical significance of such curves in the case of the equation $\displaystyle y' = x^2 + y^2$
My question is this;
The relation f(x,y)=c where c is constant, represents the family of level curves of f in the xy-plane, what is the geometrical significance of such curves in the case of the equation $\displaystyle y' = x^2 + y^2$
A level curve in the case of $\displaystyle y' = x^2 + y^2$ is a circle centered at the origin. The significance of such a curve is that $\displaystyle y'$ is constant at every point along the curve. This means that the slope of the tangent line is the same at every point along the curve. This can make it easier to draw a slope field, which can give you insight into what solutions to the differential equation look like but don't require you to actually solve the equation. In this case, I'm pretty sure the equation can't be solved by analytical means, so a slope field can be particularly useful.
You can come up with a slope field by drawing any level curve, figuring out the value of $\displaystyle y'$ for that particular level curve, and then drawing a little line segment with that slope at a bunch of points along the level curve (repeating this process for several different level curves). So for this example, you can draw a circle of any radius centered at the origin. $\displaystyle y'$ is simply the square root of the radius of the circle, and you draw little line segments with slope $\displaystyle y'$ at a bunch of points along the circle.