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# Partial Differential Equations ## Separation of Variables ### The Heat Equation $$ \frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2} $$ We look for solutions of the form $$ u(x, t) = X(x)T(t) $$ Substituting, we obtain $$ X(x)T'(t) = kX''(x)T(t) $$ Dividing both sides by...

# Partial Differential Equations ## Separation of Variables ### The Heat Equation $$ \frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2} $$ We look for solutions of the form $$ u(x, t) = X(x)T(t) $$ Substituting, we obtain $$ X(x)T'(t) = kX''(x)T(t) $$ Dividing both sides by $X(x)T(t)$, we have $$ \frac{T'(t)}{kT(t)} = \frac{X''(x)}{X(x)} $$ Since the left side depends only on $t$ and the right side depends only on $x$, both sides must be equal to a constant, say $-\lambda$. Thus, we have two ordinary differential equations: $$ \begin{aligned} T'(t) + k\lambda T(t) &= 0 \\ X''(x) + \lambda X(x) &= 0 \end{aligned} $$ ### The Wave Equation $$ \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} $$ We look for solutions of the form $$ u(x, t) = X(x)T(t) $$ Substituting, we obtain $$ X(x)T''(t) = c^2X''(x)T(t) $$ Dividing both sides by $X(x)T(t)$, we have $$ \frac{T''(t)}{c^2T(t)} = \frac{X''(x)}{X(x)} $$ Since the left side depends only on $t$ and the right side depends only on $x$, both sides must be equal to a constant, say $-\lambda$. Thus, we have two ordinary differential equations: $$ \begin{aligned} T''(t) + c^2\lambda T(t) &= 0 \\ X''(x) + \lambda X(x) &= 0 \end{aligned} $$ ### Laplace's Equation $$ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 $$ We look for solutions of the form $$ u(x, y) = X(x)Y(y) $$ Substituting, we obtain $$ X''(x)Y(y) + X(x)Y''(y) = 0 $$ Dividing both sides by $X(x)Y(y)$, we have $$ \frac{X''(x)}{X(x)} + \frac{Y''(y)}{Y(y)} = 0 $$ Thus, $$ \frac{X''(x)}{X(x)} = -\frac{Y''(y)}{Y(y)} $$ Since the left side depends only on $x$ and the right side depends only on $y$, both sides must be equal to a constant, say $-\lambda$. Thus, we have two ordinary differential equations: $$ \begin{aligned} X''(x) + \lambda X(x) &= 0 \\ Y''(y) - \lambda Y(y) &= 0 \end{aligned} $$