# Questions tagged [singular-solution]

For questions about the singular solution of the ordinary differential equations. It is a special type of solution different from general solution. Such solutions does not contain any arbitrary constant and is not a particular case of the general solution.

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### quasi-solvable ode

I face with a quasi-solvable ode $x^{2}(1+x^{2})y''+2xy'+A(1+x^{2})^{2}y=0$ where $A$ is a constant. I am trying to find a solution for this ode. My suggestion is that we can rewrite the above ...
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### Are these equations "properly" defined differential equations? (finite duration solutions to diff. eqs.)

Are these equations properly defined differential equations? Modifications were made to a deleted question to re-focus it. I am trying to find out if there exists any exact/accurate/non-approximated ...
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### Frobenius method solution

Suppose I have a second order linear differential equation. I have a solution about a regular singular point say $x=0$. Suppose the indicial equation has a repeated root for the indicial equation. I ...
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### Is $r(t) = \frac{1}{144}(T-t)^4\theta(T-t)$ a valid solution to $\ddot{r}=\sqrt{r},\,r(0)=\frac{T^4}{144}>0$? (with $\theta(t)$ the Heaviside step fn)

Is $r(t) = \frac{1}{144}(T-t)^4\theta(T-t)$ a valid solution to $\ddot{r}=\sqrt{r},\,r(0)=\frac{T^4}{144}>0$? (with $\theta(t)$ the Heaviside unitary step function) I am looking here for examples ...
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### Singular solution of the differential equation $(y')^2-3xy'+y^2=0$.

To find singular solution of the differential equation $$(y')^2-3xy'+y^2=0$$ I find its $p$-discriminant as $$y=\pm \frac {3}{2}x$$ but none of the factor satisfies differential equation. How to ...
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### Studying singular solutions of the ODE

Given that $$8x(y')^3 + 12y(y')^2 - 9y^5 = 0$$ Find all solutions and study singular solutions. My solution After noting that $y' = 0$ is solution only when $y = 0$ we subtitute $y' = \frac{1}{x'}$ ...
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So I'm trying to do this proof, The form of Clairauts equation is $$y(x) = xy' + f(y')$$ You differentiate once to get $$y' = y' + xy'' + f'(y')y''$$ You rearrange and get two solutions The general ...