(y')^2+y^2=-2 why this equation has no general solution ?

In summary, this differential equation has no general solution because it is non-linear and the left-hand side is necessarily non-negative while the right-hand side is negative. Non-linear equations rarely have general solutions and the only possible solutions in this case would involve complex numbers, making it difficult to find a general solution in terms of real valued functions.
  • #1
young_eng
8
0
hi

(y')^2+y^2=-2
why this differential equation has no general solution ?
 
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  • #2
It's non-linear. Having a general solution for these types of equations is the exception, not the rule.

Not that there isn't a general solution because I don't know. It's just that non-linear equations rarely have general solutions
 
  • #3
Because the lhs is necessarily non-negative (sum of squares), whereas the lhs is negative. You can have a solution in complex numbers.
 
  • #4
young_eng said:
hi

(y')^2+y^2=-2
why this differential equation has no general solution ?

Why not just solve it the regular way:

[tex]\frac{dy}{\sqrt{-2-y^2}}=\pm dx[/tex]

or [itex]y=\pm i\sqrt{2}[/tex] are solutions, maybe singular ones. Not sure. Otherwise:

[tex]\frac{y\sqrt{-2-y^2}}{2+y^2}=\tan(c\pm x)[/tex]

[tex]y(x)=\pm \frac{\sqrt{2}\tan(c\pm x)}{\sqrt{-\sec^2(c\pm x)}}[/tex]

so that the solution is in the form of y(z)=u+iv
 
  • #5
There is no general solution in terms of real valued functions because if y' and y are both real numbers (for a given x) then [itex](y')^2+ y^2[/itex] cannot be negative!

Oops! Dickfore had already said that, hadn't he?
 
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Related to (y')^2+y^2=-2 why this equation has no general solution ?

1. Why is this equation unsolvable?

This equation is unsolvable because it does not have a general solution that can be expressed in terms of standard mathematical operations and functions. This means that there is no set of values that can be substituted for y and y' that will satisfy the equation.

2. Can this equation be solved using advanced mathematical methods?

No, this equation cannot be solved using advanced mathematical methods such as calculus or differential equations. These methods require the equation to have a general solution, which is not the case for this equation.

3. Is there any specific condition under which this equation can have a solution?

No, there is no specific condition or set of conditions that can be applied to this equation to make it solvable. The lack of a general solution is a fundamental property of this equation.

4. Are there any real-life applications of this equation?

This equation is an example of a non-linear differential equation, which is commonly used in physics and engineering to model real-world phenomena. However, in its current form, it does not have a general solution and therefore cannot be directly applied in practical situations.

5. Can this equation be rewritten or manipulated to have a solution?

No, this equation cannot be rewritten or manipulated to have a general solution. Any attempts to do so would result in an equation that is fundamentally different from the original and would not accurately represent the same problem.

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