Nonlinear ODE initial value problemA student came to me with a problem I couldn't solve. It's the beginning of the semester in his Intro DiffEq class, and so the solution shouldn't be too difficult. But it completely stumped me, and now I can't let it go! Here it is:Problem. Find all solutions to the IVP:(dydx)2−4x2=x4−y2,;;;;y(0)=0.I thought about maybe just doing a sort of guess-and-check method, but the existence/uniqueness theorem doesn't apply, so I'm not sure that would help even if I could find a single solution, which I cannot in any case.

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Nonlinear ODE initial value problemA student came to me with a problem I couldn&#039;t solve. It&#039;s the beginning of the semester in his Intro DiffEq class, and so the solution shouldn&#039;t be too difficult. But it completely stumped me, and now I can&#039;t let it go! Here it is:Problem. Find all solutions to the IVP:(dydx)2−4x2=x4−y2,;;;;y(0)=0.I thought about maybe just doing a sort of guess-and-check method, but the existence/uniqueness theorem doesn&#039;t apply, so I&#039;m not sure that would help even if I could find a single solution, which I cannot in any case.

j3jell5 · Accepted Answer

Step 1Keep in mind that, by the statement of the equation,&amp;nbsp;y′(x)2≥4x2&amp;nbsp;and&amp;nbsp;x2≥y(x)2. The former restriction is equivalent to&amp;nbsp;y′(x)≥2x&amp;nbsp;or&amp;nbsp;y′(x)≤−2x. The latter restriction is equivalent to&amp;nbsp;-x2≤y(x)≤x2. Therefore,y′(x)=±x4+4x2−y(x)2with the restrictions already specified. These are two equations of the formy′(x)=f[x,y(x)],where&amp;nbsp;f:R2→R. Notice that if&amp;nbsp;y(x)=x2, thenStep 2x4+4x2−y(x)2=2|x|,hence&amp;nbsp;y(x)=x2&amp;nbsp;solves the former equation for&amp;nbsp;x&amp;gt;0, and it solves the latter for&amp;nbsp;x&amp;lt;0. Meanwhile,&amp;nbsp;y(x)=−x2&amp;nbsp;solves the former equation for&amp;nbsp;x&amp;lt;0, and solves the latter for&amp;nbsp;x&amp;gt;0. Thus, the former equation is solved by&amp;nbsp;y(x)=x|x|&amp;nbsp;on&amp;nbsp;R&amp;nbsp;and the latter is solved by&amp;nbsp;y(x)=−x|x|&amp;nbsp;on&amp;nbsp;R. With this in mind, you should try&amp;nbsp;z(x)=|x|y(x), and work with this, to find other solutions.

Nonlinear ODE initial value problem A student came to me with a problem I couldn't solve. It's t

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