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I'm trying to solve the following ODE asymptotically.

$$y(x)^2 y'(x)^2-\left(\sqrt{2} x\right)^2 y'(x)^2+y(x)^2=0$$

From

AsymptoticDSolveValue[{y[x]^2*y'[x]^2 + y[x]^2 - ( x*Sqrt[2])^2*y'[x]^2 == 0},
                      y[x], {x, 0, 10}]

I get some complex valued stuff

 -I x - (138 x^10)/(25 C[1]^9) + (857 I x^9)/(3240 C[1]^8) - (73 x^8)/(
 40 C[1]^7) + (3 I x^7)/(2 C[1]^6) + (4 x^6)/(9 C[1]^5) + (3 I x^5)/(
 10 C[1]^4) + x^4/(2 C[1]^3) - (I x^3)/(3 C[1]^2) + C[1]

However, one can easily check, that $y(x)=x$ solves the equation above. I also get the error message AsymptoticDSolveValue::asdb:

error

My question is:

How do I get all the branches of the solution from AsymptoticDSolveValue[], or if that is impossible, how do I impose that the desired solution is real?

EDIT: The answer by Nasser adresses part of my question, however it doesn't provide an explanation on whether or not it is possible to obtain multiple branches from AsymptoticDSolveValue[]

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1 Answer 1

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DSolve does not find $y(x)=x$ either, and I think this is why AsymptoticDSolveValue does not. I am sure they share some core code internally.

DSolve returns general solutions, but they are implicit. But it does not find $y=x$

Mathematica graphics

btw, Maple does finds $y=x$, and if you use its option singsol=all it will also return $y=0$ solution (singular solution).

It would be nice if DSolve could have an explicit option to return singular solutions to ode's (if they exist), in addition to the general solution.

restart;
ode:=y(x)^2*diff(y(x),x)^2+y(x)^2-(x*sqrt(2))^2*diff(y(x),x)=0
dsolve(ode,singsol=all)

Mathematica graphics

May be you could send a report on this to WRI support asking why $y=x$ was missed. You can include

ClearAll[y, x];
ode = y[x]^2*y'[x]^2 + y[x]^2 - (x*Sqrt[2])^2*y'[x]^2 == 0;
ode /. y -> Function[{x}, x]

Mathematica graphics

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  • $\begingroup$ This particular ODE is really difficult to analyse with Mathematica. In general I am aiming to solve for a general form, where $\sqrt{2}x$ is replaced with some $r(x)$. Even though I managed to obtain some exact, real solutions for various $r(x)$, Mathematica almost always spits out only complex valued functions for both DSolve and AsymptoticDSolveValue. Thanks, for the Maple output, I will see what I can do with that programme. $\endgroup$ May 11, 2020 at 2:46
  • $\begingroup$ Also, can you impose only real solutions in DSolve? $\endgroup$ May 11, 2020 at 3:12
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    $\begingroup$ @MichałKuczynski can you impose only real solutions in DSolve, No. Not that I know about. DSolve returns the solutions it find, then one can filter out real or complex after that if they want. $\endgroup$
    – Nasser
    May 11, 2020 at 4:10

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