Problem

Prove that the initial-value problem $$ y^{\prime…

02:08

Problem 12 Easy Difficulty

Verify that the given function (or relation) defines a solution to the given differential equation and sketch some of the solution curves. If an initial condition is given, label the solution curve corresponding to the resulting unique solution. (In these problems, $c$ denotes an arbitrary constant.)
$$(x-c)^{2}+y^{2}=c^{2}, \quad y^{\prime}=\frac{y^{2}-x^{2}}{2 x y}, \quad y(2)=2$$

Answer

$(x-2)^{2}+y^{2}=2^{2}$

Related Courses

Calculus 2 / BC

Differential Equations and Linear Algebra

Chapter 1

First-Order Differential Equations

Section 3

The Geometry of First-Order Differential Equations

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Video Transcript

All right. So where this is a set of curves weaken, Graph it. It's actually a set of circles. They're old centred on points on the X access where y equals zero on me all they'll pass through the origin. So it looked like this. Uh uh. Casualties are more strange. Smaller circle. There's a circle in between. There's another circle here, once a on on the other side. We have the same thing and we check that this is a solution to the differential equation. So we we want, you know, way. Want to differentiate Mr Question two X minus C close. Your eyes were differentiating this equation with respect to acts, and this is what we cats Now what we already know about this is that well, we can simplify this equation. His ex quit. So that's what's nervous. Down a little bit. We can simplify this equation. Quite a close. Your squared. This is X squared minus your accident plus two square plus y square close to the square. These terms cancel. Are we good? C equals exporting plus y squared over two acts an AK minus simply because x squared minus y squared over Truax because This is a two X squared term up here. And so we have to times this thing X squared minus y ask wine over and two X plus two y y prime equals zero. And if we solve this got wired primes equals y squared minus x square over to sex. Why, This is an application of physically complacent differentiation and that shows that the solution these current, this family of curves are all solutions to Mr Potential equation.

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