How to determine the integration constants in solving the Klein Gordon equation?

I solved by

##
\int d \dot{\phi} = \int d x \to
\dot{\phi} = x+ c_1 \to \int d \phi = \int d t ( x+c_1)
\to \phi = x t + c_1 t + c_2
##

Is this way correct? To determine ##c_2## use initial condition: ##\phi(0,x)=0## that yields ##c_2=0##, but how to get ##c_1## ?

The kernel of the operator [itex]\partial_x[/itex] consists of all functions [itex]f[/itex] for which [itex]\partial_x f = 0[/itex]; this includes constants, but also includes functions which depend only on [itex]t[/itex]. Therefore the most general element of this kernel is [itex]A(t)[/itex]. Hence [tex]\int \partial_x \dot \phi\,dx = \int 0\,dx \Rightarrow \dot\phi = A(t).[/tex] Now integrate with respect to [itex]t[/itex]. This time, the most general element of the kernel of [itex]\partial/\partial t[/itex] is a function of [itex]x[/itex] alone.

Safinaz said:

Homework Statement: How to solve the following wave equation for the scalar $\phi(t,x)$ :
Relevant Equations: ##\partial_i \dot{\phi}=0 ##

Where ##\partial_i= \partial/\partial x##. And (.) is the derivative with respect for time ## \partial/\partial t##

I solved by

##
\int d \dot{\phi} = \int d x \to
\dot{\phi} = x+ c_1 \to \int d \phi = \int d t ( x+c_1)
\to \phi = x t + c_1 t + c_2
##

Is this way correct? To determine ##c_2## use initial condition: ##\phi(0,x)=0## that yields ##c_2=0##, but how to get ##c_1## ?

Your function provably does not satisfy ##\partial_i\partial_t \phi = 0##. Just try to differentiate it!

(apart from what was already said)

pasmith said:

The kernel of the operator [itex]\partial_x[/itex] consists of all functions [itex]f[/itex] for which [itex]\partial_x f = 0[/itex]; this includes constants, but also includes functions which depend only on [itex]t[/itex]. Therefore the most general element of this kernel is [itex]A(t)[/itex]. Hence [tex]\int \partial_x \dot \phi\,dx = \int 0\,dx \Rightarrow \dot\phi = A(t).[/tex] Now integrate with respect to [itex]t[/itex]. This time, the most general element of the kernel of [itex]\partial/\partial t[/itex] is a function of [itex]x[/itex] alone.

You mean

##\dot{\phi} =A(t) \to \int \partial_t \phi = \int A(t) dt \to \phi = A(t) t + c ? ##
but what is A(t) ?

Safinaz said:

You mean

##\dot{\phi} =A(t) \to \int \partial_t \phi = \int A(t) dt \to \phi = A(t) t + c ? ##
but what is A(t) ?

First of all, you cannot integrate a general function A(t) with respect to t and obtain A(t) t. Not even in single variable calculus.

Second, you are still missing what was said. If ##\partial_t \phi = A(t)##, then ##\phi = a(t) + f(x)##, where ##a’(t) = A(t)##. The “integration constant” when integrating a partial derivative is generally a function of all of the other variables.

Orodruin said:

First of all, you cannot integrate a general function A(t) with respect to t and obtain A(t) t. Not even in single variable calculus.

Second, you are still missing what was said. If ##\partial_t \phi = A(t)##, then ##\phi = a(t) + f(x)##, where ##a’(t) = A(t)##. The “integration constant” when integrating a partial derivative is generally a function of all of the other variables.

Okay. But now how to get the definition of ##f(x)## and ##a(t)## ?

Safinaz said:

Okay. But now how to get the definition of ##f(x)## and ##a(t)## ?

Just as you need boundary or initial conditions to fix integration constants for ODEs, you will need boundary/initial conditions to fix those functions.

Orodruin said:

Just as you need boundary or initial conditions to fix integration constants for ODEs, you will need boundary/initial conditions to fix those functions.

Hello. Thanks so much for your answer. I was trying to find proper IC and BC to find ## \phi(t,x)## . Assuming:

##
bc={\phi[t,0]==1,(D[\phi[t,x],x]/.x->Pi)==0}
##
##
ic={\phi[0,x]==0,(D[\phi[t,x],t]/.t->0)==1}
##

Also ## \phi(t,x)## obays the Klein Gordon’s equation :
## \left( \frac{\partial^2}{\partial t^2} - \frac{\partial^2}{\partial x^2} \right) \phi = 0 ##

in which solution:
##
\phi(t,x) = ( c_1 e^{kt} + c_2 e^{-kt} ) ( c_3 e^{kx} + c_4 e^{-kx} ) …………(1)
##

To find the constants in Eq. (1) , BC leads to ##c_3 = c_4= 1/2 ## and the IC leads to to ##c_1=- c_2= 1/2 ## is that correct? But how to know ## k ## ?