Question #20671

look at the partial differential equation
xux - tut= u
show that u( x , t) = xf( xt) is a general solution , where f is any differentiable function

Expert's answer

X, T - variable

()x, ()t, ()' - derivative

X*Ux - T*Ut = U

U (X, T) = X* F (X*T)

X* (X* F (X*T))x - T* (X* F (X*T))t = X* F (X*T)

X* (F (X*T) + X*F'(X*T)*T) - T* (X* F'(X*T)* X) = X* F (X*T)

X*F (X*T) +{X^2*F'(X*T)*T - T* X^2* F'(X*T)} = X* F (X*T)

X*F (X*T)=X*F (X*T)

()x, ()t, ()' - derivative

X*Ux - T*Ut = U

U (X, T) = X* F (X*T)

X* (X* F (X*T))x - T* (X* F (X*T))t = X* F (X*T)

X* (F (X*T) + X*F'(X*T)*T) - T* (X* F'(X*T)* X) = X* F (X*T)

X*F (X*T) +{X^2*F'(X*T)*T - T* X^2* F'(X*T)} = X* F (X*T)

X*F (X*T)=X*F (X*T)

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