Find the singular points of the ODE

Question

Consider the ODE $$\frac{d^2x}{dt^2}+e^t\frac{dx}{dt}+\frac{x}{1+2t}=0$$ What are the singular points of this ODE?

For an ODE of the form $$P(t)\frac{d^2x}{dt^2}+Q(t)\frac{dx}{dt}+R(t)x=0$$ a point is singular if $P,Q,R$ are analytic and $P(t_0)=0$. From the equation above, there doesn't seem to be any singular points. We have that $P(t)=1$ which does not equal $0$ for any value $t_0$.

The reason I asked is because when solving this using the series method with $x(t)=\sum^\infty_{n=0}a_nt^n$, I want to know at what values $t$ will the solution not converge. I know it won't converge at singular points, but there doesn't seem to be any.

Answer 1

You are using the incorrect form of the ODE, instead rewrite it as $$(1+2t)\frac{d^2x}{dt^2} + (1+2t)e^t\frac{dx}{dt} + x = 0$$ so that $$P(t) = 1 + 2t.$$ Thus you have a singular point at $t = -1/2$.

Answer 2

We don't need re-write the ODE, just notice that $$\frac{{\rm d}^{2}x}{{\rm d}t^{2}}+e^{t}\frac{{\rm d}x}{{\rm d}t}+\frac{1}{1+2t}x=0$$ Hence the ODE has the form $$\frac{{\rm d}^{2}x}{{\rm d}t^{2}}+P(t)\frac{{\rm d}x}{{\rm d}t}+Q(t)x=0$$ Now,

• A point $t_{0}$ is ordinary point if $P(t)$ and $Q(t)$ are analytic at $t_{0}$.

• A point $t_{0}$ is singular point if $P(t)$ or $Q(t)$ is not analytic at $t_{0}$.

• A point $t_{0}$ which is singular point is regular singular point if $(t-t_{0})P(t)$ and $(t-t_{0})^{2}Q(t)$ are analytics at $t_{0}$ otherwise the point $t_{0}$ is an irregular singular point.

Setting $P(t)=e^{t}$ and $Q(t)=\frac{1}{1+2t}$ so since $1+2t=0\implies t=-1/2$ so $Q(t)$ is not analytic in $t_{0}=-1/2$.

Now, we need to know if that point is regular or irregular singular point. For that part notice $(t+1/2)e^{t}$ is analytic at $t=-1/2$ but in $\displaystyle (t+1/2)^{2}\cdot \frac{1}{1+2t}$ we can see is not defined at $t=-1/2$ but we can still see that this singularity is weak because $\displaystyle \lim_{t\to -1/2}(t+1/2)^{2}\cdot\frac{1}{1+2t}=0$ which is finite.

Therefore all work and we can conclude $t_{0}=-1/2$ is a regular singular point and then we can use the method of Frobenius for to solve the ODE.

Just a small remark:

• I think you should change the notation $x(t)$ to $y(x)$ since for the general $x(t)$ describes the position relative to the temporal variable and here it is strange to have $t=-1/2$. I think the second notation would be more convenient.