Can You Solve This Non-linear PDE with Variable Separation?

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The discussion focuses on solving the non-linear partial differential equation (PDE) defined by U^2_xU_t - 1 = 0 with the initial condition U(x, 0) = x. The equation allows for separation of variables, leading to the solution format U(x,t) = f(x) + g(t). By substituting into the equation, it is established that f'(x)^2 = 1/g'(t) = constant, ultimately confirming the solution as U = x + t.

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aminfar
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I am new to non-linear PDEs. So I tried to solve it, but I stuck in the beginning.

U^2_xU_t - 1 = 0

U(x, 0) = x
 
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The equation admits a separation of variables, and the solution can be written as U(x,t)=f(x)+g(t). Inserting this, you get f'(x)^2=1/g'(t)=constant. The solution can then be checked to be U=x+t
 

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