# Search results

1. ### Potential series method

I think that in the case when \alpha(x)y''(x)+\beta(x)y'(x)+\gamma(x)y(x)=0 if ##\alpha(0)=0## you must work with ##\sum^{\infty}_{n=0}a_nx^{n+k}##, but I'm not sure.
2. ### Potential series method

Why sometimes we search solution of power series in the way: y(x)=\sum^{\infty}_{n=0}a_nx^n and sometimes y(x)=\sum^{\infty}_{n=0}a_nx^{n+1}???
3. ### Coefficients derivative

But what if for some ##x##, ##m(x)=0##.
4. ### Coefficients derivative

Why we always write equation in form y''(x)+a(x)y'(x)+b(x)=f(x) Why we never write: m(x)y''(x)+a(x)y'(x)+b(x)=f(x) Why we never write coefficient ##m(x)## for example?
5. ### Picard method of succesive approximation

And what is X?
6. ### Green function as distributions

Can you help me?
7. ### Green functions

Can you help me?
8. ### Green function as distributions

To be more precise. If I say solution to eq is u(x)=\int^{x}_0g(x,s)f(s)ds where g=e^{-\int^x_yp(z)dz} Then if I define g=H(x-s)e^{-\int^x_sp(z)dz} is then u(x)=\int^{\infty}_0g(x,s)f(s)ds and what is Green function this g=e^{-\int^x_yp(z)dz} or this...
9. ### Green functions

Tnx. Just to ask you how I get this boundaries in integration? u_h'+p(x)u_h=0 \frac{du_h}{dx}=-p(x)u_h(x) u_h=Ce^{-\int p(x)dx} So u(x)=C(x)e^{-\int p(x)dx} u'=C'(x)e^{-\int p(x)dx}-pu When I put this to Eq u'+p(x)u=f(x) I get C'(x)e^{-\int p(x)dx}-pu+pu=f and...
10. ### Green function as distributions

If we have Green function g(x,s)=exp[-\int^x_s p(z)dz] we want to think about that as distribution so we multiply it with Heaviside step function g(x,s)=H(x-s)exp[-\int^x_s p(z)dz] Why we can just multiply it with step function and tell that the function is the same. Tnx for the answer.
11. ### Green functions

Eq u'(x)+p(x)u=f(x) with initial condition u(0)=0 It's homogenous solution is u_h=Ce^{-\int^x_0 p(s)ds} Complete solution u(x)=e^{-\int^x_0 p(s)ds}\int^x_0f(s)e^{\int^s_0 p(z)dz}ds=\int^x_0 f(s)g(x,s)ds where g(x,s)=e^{-\int^{x}_s p(\xi)d \xi } I didn't see that last...
12. ### Picard method of succesive approximation

y(t) = y0 + \int^{ t}_{t_0} f(s, y(s)) ds. Picard’s method starts with the deﬁnition of what it means to be a solution: if you guess that a function φ(t) is a solution, then you can check your guess by substituting it into the right-hand side of equation (2) and comparing it to the...