Thread: Partial Differential eqn - method of characteristics

Am going through problems from book and have come across a section of my lecture notes that i don't understand, i'm sure it's something small i'm missing but can't get my head around it.

In lecture notes we solve the wave eqn;

it gets to the General solution which is;

Q(x,t) = F(x-t) +G(x-t)

and then as one of my initial conditions is;

dQ/dx(0,t) = 2 (where this is the partial derivative of Q wrt. x)

i need to find an expression for the dQ/dx, which was written as;

dQ/dx = F'(x+t) + G'(x-t)
i dont understand how this works?
can anyone help?

Originally Posted by monster

Am going through problems from book and have come across a section of my lecture notes that i don't understand, i'm sure it's something small i'm missing but can't get my head around it.

In lecture notes we solve the wave eqn;

it gets to the General solution which is;

Q(x,t) = F(x-t) +G(x-t)

and then as one of my initial conditions is;

dQ/dx(0,t) = 2 (where this is the partial derivative of Q wrt. x)

i need to find an expression for the dQ/dx, which was written as;

dQ/dx = F'(x+t) + G'(x-t)
i dont understand how this works?
can anyone help?

It's the chain rule. Let and so so

Consider the IVP

Parameterizing your variable as and assuming that you start on a characteristic curve with initial conditions then the solution to the IVP can be found by the ODE