Deciphering Confusing Differential Operator Problems

[ELESSAR TELKONT]

I have two problems and I don't know what they want to tell. Please tell me what do you think

1. We define operator $$L[x]=a(t)\ddot{x}+b(t)\dot{x}+c(t)x$$ in $$C^{2}(I)$$ function space. Proof that $$\frac{\partial}{\partial\lambda}L[x]=L\left[\frac{\partial x}{\partial\lambda}\right]$$. ¿What do you think the lambda is for? I don't understand! We haven't done anything like that in the course.

2.Bessel equation of zero order. Use the Frobenius Method to show that $$L[x]=a_{0}\lambda^{2}t^{\lambda}$$, with the supposition that the coefficient of $$t^{n+\lambda}$$ for $$n\geq 1$$ vanishes and that the root of the indical polinomial is of multiplicity 2, and show that $$L\left[\frac{\partial x}{\partial\lambda}\right]=2a_{0}\lambda t^{\lambda}+a_{0}\lambda^{2}t^{\lambda}\ln t$$. ¿What do you think the lambda is for? I have searched in books and internet and I never saw that the Bessel equation of zero order have the form that this problem makes use.

Please help me to decipher what the hell teacher's assistant was thinking when he wrote the homework. It's urgent.

 

[HallsofIvy]

I have two problems and I don't know what they want to tell. Please tell me what do you think

1. We define operator $$L[x]=a(t)\ddot{x}+b(t)\dot{x}+c(t)x$$ in $$C^{2}(I)$$ function space. Proof that $$\frac{\partial}{\partial\lambda}L[x]=L\left[\frac{\partial x}{\partial\lambda}\right]$$. ¿What do you think the lambda is for? I don't understand! We haven't done anything like that in the course.

You have every right to ask! I suspect the "$\lambda$" was supposed to be "t" since the coefficients in L depend on t. Or the other way around. In any case, the differentiation is with respect to the parameter.

2.Bessel equation of zero order. Use the Frobenius Method to show that $$L[x]=a_{0}\lambda^{2}t^{\lambda}$$, with the supposition that the coefficient of $$t^{n+\lambda}$$ for $$n\geq 1$$ vanishes and that the root of the indical polinomial is of multiplicity 2, and show that $$L\left[\frac{\partial x}{\partial\lambda}\right]=2a_{0}\lambda t^{\lambda}+a_{0}\lambda^{2}t^{\lambda}\ln t$$. ¿What do you think the lambda is for? I have searched in books and internet and I never saw that the Bessel equation of zero order have the form that this problem makes use.

Here, it is clear. The parameter [itex]\lambda[\itex] appears in the formula itself. As far as that being "Bessel's equation", it really doesn't matter. Just use Frobenious' method to solve that differential equation for every $\lambda$.<br /> <br /> Please help me to decipher what the hell teacher's assistant was thinking when he wrote the homework. It's urgent.[/QUOTE][/itex]