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Partial-Differential-Equations · Pde-Solutions

Does u = exp(-t)*sin(x) satisfy u_t = u_xx

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

Step by step

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

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

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

    Differentiate with respect to t, holding the other variables constant.

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

    Differentiate 2 times with respect to x.

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

    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^{- t} \sin{\left(x \right)} \text{ satisfies } u_{t} = u_{xx}