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Partial-Differential-Equations · Pde-Solutions

Does u = exp(x)*cos(y) satisfy u_xx + u_yy = 0

u = e^{x} \cos{\left(y \right)},\quad u_{xx} + u_{yy} = 0

Step by step

  1. u = e^{x} \cos{\left(y \right)},\qquad u_{xx} + u_{yy} = 0

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

  2. u_{xx} = e^{x} \cos{\left(y \right)}

    Differentiate 2 times with respect to x.

  3. u_{yy} = - e^{x} \cos{\left(y \right)}

    Differentiate 2 times with respect to y.

  4. \text{LHS} = 0,\quad \text{RHS} = 0

    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^{x} \cos{\left(y \right)} \text{ satisfies } u_{xx} + u_{yy} = 0