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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} = - e^{2 x} \cos^{2}{\left(y \right)},\quad \text{RHS} = 0

    Substitute the derivatives into both sides.

  5. \text{LHS} - \text{RHS} = - e^{2 x} \cos^{2}{\left(y \right)}

    Subtract and simplify.

  6. \text{LHS} - \text{RHS} = - e^{2 x} \cos^{2}{\left(y \right)} \neq 0

    The residual is not identically zero, so this function does not solve the equation.

Reveal the answer
\text{No: } \text{LHS} - \text{RHS} = - e^{2 x} \cos^{2}{\left(y \right)}