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    I What are partial differential equations?

    The answer to your question is true. For example, U(x,t) is a function that depends on x and t. X and t are independent. Do you think is possible for a dependent function to have derivatives as functions?
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    I 2nd Order Linear ODE w/ nonconstant coefficient

    The answer in wolfram alpha is written in a compact form using the substitution e^2x. If the left hand side of the equation is solved implicitly you should get a answer to that portion of their answer that will look similar to the equation that you started with
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