Given an equation $F(u,v)=0$ where $u$ and $v$ are functions of $x,y,z$.

Differentiate $F(u,v)=0$ with respect to $x$ and $y$.

Eliminate the function $F$ from the resulting equations.

Integrate with respect to the variable indicated by the partial derivative, treating other variables as constants. Remember to add an arbitrary function of the "constant" variable(s) instead of a constant.
Example: If $\frac{\partial^2 z}{\partial x \partial y} = f(x,y)$, integrate with respect to $y$ first, then $x$.

For PDEs of the form $Pp + Qq = R$ (where $P, Q, R$ are functions of $x,y,z$):
1. Write the auxiliary equations: $\frac{dx}{P} = \frac{dy}{Q} = \frac{dz}{R}$.
2. Solve these ordinary differential equations to find two independent solutions, $u(x,y,z) = a$ and $v(x,y,z) = b$.
3. The general solution is given by $\Phi(u,v) = 0$ or $u = \phi(v)$.

Assume the solution $u(x,t)$ can be written as a product of functions of single variables, e.g., $u(x,t) = X(x)T(t)$.
Substitute this into the PDE.
Separate the variables such that one side depends only on $x$ and the other only on $t$.
Equate both sides to a constant (separation constant, e.g., $\lambda$).
Solve the resulting two ordinary differential equations for $X(x)$ and $T(t)$.
Combine $X(x)$ and $T(t)$ to get the general solution.
Use initial and boundary conditions to find the constants.

1D Wave Equation: $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$
- Solution form: $u(x,t) = (C_1 \cos(kx) + C_2 \sin(kx))(C_3 \cos(ckt) + C_4 \sin(ckt))$
1D Heat Equation: $\frac{\partial u}{\partial t} = c^2 \frac{\partial^2 u}{\partial x^2}$
- Solution form: $u(x,t) = e^{-c^2 k^2 t} (C_1 \cos(kx) + C_2 \sin(kx))$

For a function $f(x)$ defined on $(-\infty, \infty)$: $$ f(x) = \frac{1}{\pi} \int_0^\infty \int_{-\infty}^\infty f(t) \cos(\lambda(t-x)) dt d\lambda $$
Alternatively: $$ f(x) = \int_0^\infty [A(\lambda) \cos(\lambda x) + B(\lambda) \sin(\lambda x)] d\lambda $$ where $A(\lambda) = \frac{1}{\pi} \int_{-\infty}^\infty f(t) \cos(\lambda t) dt$ and $B(\lambda) = \frac{1}{\pi} \int_{-\infty}^\infty f(t) \sin(\lambda t) dt$.

If $f(x)$ is an even function: $$ f(x) = \int_0^\infty A(\lambda) \cos(\lambda x) d\lambda $$ where $A(\lambda) = \frac{2}{\pi} \int_0^\infty f(t) \cos(\lambda t) dt$.
If $f(x)$ is an odd function: $$ f(x) = \int_0^\infty B(\lambda) \sin(\lambda x) d\lambda $$ where $B(\lambda) = \frac{2}{\pi} \int_0^\infty f(t) \sin(\lambda t) dt$.

Full Fourier Transform: $\mathcal{F}\{f(x)\} = F(s) = \int_{-\infty}^\infty f(x) e^{isx} dx$
Fourier Sine Transform: $\mathcal{F}_s\{f(x)\} = F_s(s) = \int_0^\infty f(x) \sin(sx) dx$
Fourier Cosine