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solve Uxx-3Uxt-4Utt=0 with u(x,0)=x^2 and Ut(x,0)=e^x

I know that this is hyperbolic since D=(-1.5)^2+4 >0 so I have to transform the variables x and t linearly to obtain the wave equation of the form

(Utt-c^2Uxx=0). The above equation is equivalent to:

(d/dx - 1.5 d/dt)*(d/dx - 1.5 d/dt)u - 6.25 d^2u/dt^2 = 0

let x=b

let t=-1.5b + 2.5a

Thus,

Ub=Ux - (1.5) Ut

Ua=2.5 Ut

thus Ubb-Uaa=0. This is where I am stuck..

I know the general solution is u(a,b)=f(a+b)+g(a-b)

also the explicit solution is u(a,b)=(1/2)*[φ(a+b)+φ(a-b)]*(1/2c)*(integral

ψ(s)ds from a-b to a+b).

where u(a,0)=φ(a) and Ub(a,0)=ψ(a).

The solution is (4/5)*[e^(x+t/4)-e^(x-t)]+x^2+(1/4)*t^2

but how to obtain it?

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# Solve Uxx-3Uxt-4Utt=0 (hyperbolic)

**Physics Forums | Science Articles, Homework Help, Discussion**