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Partial-Differential-Equations · Pde-Solutions

Does u = exp(-(x+t)^2) satisfy u_tt = u_xx

u = e^{- \left(t + x\right)^{2}},\quad u_{tt} = u_{xx}

Step by step

  1. u = e^{- \left(t + x\right)^{2}},\qquad u_{tt} = u_{xx}

    To check a solution, compute every derivative the equation uses, substitute, and see whether both sides agree.

  2. u_{tt} = 2 \left(2 \left(t + x\right)^{2} - 1\right) e^{- \left(t + x\right)^{2}}

    Differentiate 2 times with respect to t.

  3. u_{xx} = 2 \left(2 \left(t + x\right)^{2} - 1\right) e^{- \left(t + x\right)^{2}}

    Differentiate 2 times with respect to x.

  4. \text{LHS} = 2 \left(2 \left(t + x\right)^{2} - 1\right) e^{- \left(t + x\right)^{2}},\quad \text{RHS} = 2 \left(2 \left(t + x\right)^{2} - 1\right) e^{- \left(t + x\right)^{2}}

    Substitute the derivatives into both sides.

  5. \text{LHS} - \text{RHS} = 0

    Subtract and simplify.

  6. \text{LHS} - \text{RHS} = 0\ \checkmark

    The two sides agree for every value of the variables, so it is a solution.

Reveal the answer
\text{Yes: } u = e^{- \left(t + x\right)^{2}} \text{ satisfies } u_{tt} = u_{xx}