## List of Differential Equations Formulas Latex Code

rockingdingo 2023-01-28 #math #differential equations #laplace #bessel 2 0

List of Differential Equations Formulas Latex Code

In this blog, we will summarize the latex code for differential equations formulas, including diffusion (conduction) equation, wave equation, heat equation, laplace equation, Legendre's equation, Bessel's equation, Spherical Harmonics Equation, etc.

### 1. Differential Equations

• #### 1.0 Diffusion (conduction) Equation

##### Latex Code
            \kappa u_{xx} = u_{t} \\
\frac{\partial \Psi}{\partial t}=\kappa \triangledown^{2} \Psi

##### Explanation

• : First Order Partial Derivative of function u(x,t) over time t.
• : Second Order Partial Derivative of function u(x,t) over x.

• #### 1.1 Wave Equation

##### Latex Code
            u_{tt}=c^{2}u_{xx}

##### Explanation

• : Second Order Partial Derivative of function u(x,t) over time t.
• : Second Order Partial Derivative of function u(x,t) over x.

• #### 1.2 Heat Equation

##### Latex Code
            u_{t}={\alpha}^{2}u_{xx}

##### Explanation

• : First Order Partial Derivative of function u(x,t) over time t.
• : Second Order Partial Derivative of function u(x,t) over x.

• #### 1.3 Laplace Equation

##### Latex Code
            u_{xx} = 0

##### Explanation

• : Second Order Partial Derivative of function u(x,t) over x.

• #### 1.4 Legendre's Equation

##### Latex Code
            (1-x^{2})\frac{\mathrm{d}^{2} y}{\mathrm{d} x^{2}}-2x\frac{\mathrm{d} y}{\mathrm{d} x}+l(l+1)y=0 \\
P_{l}(x)=\frac{1}{2^{l}l!}(\frac{\mathrm{d}}{\mathrm{d} x})^{l}(x^2-1)^{l}\\
P_{l}(x)=\frac{1}{l}[(2l-1)xP_{l-1}(x)-(l-1)P_{l-2}(x)]

##### Explanation

• Solutions of Legendre equations are Legendre polynomials
• Recursion relation:

• #### 1.5 Bessel's equation

##### Latex Code
            x^{2}\frac{\mathrm{d}^{2} y}{\mathrm{d} x^{2}}+x\frac{\mathrm{d} y}{\mathrm{d} x}+(x^{2}-m^{2})y=0 \\
J_{m}(x)=\sum^{\infty}_{k=0}\frac{(-1)^{k} (x/2)^{m+2k}}{k!(m+k)!}

##### Explanation

• Solutions of Bessel equations are Bessel functions .

• #### 1.6 Spherical Harmonics Equation

##### Latex Code
            [\frac{1}{\sin \theta} \frac{\partial}{\partial \theta}(\sin \theta \frac{\partial}{\partial \theta}) + \frac{1}{\sin^{2} \theta} \frac{\partial^{2}}{\partial \phi^{2}}) ] Y^{m}_{l} + l(l+1) Y^{m}_{l}=0 \\ Y^{m}_{l}(\theta,\phi)=\sqrt{\frac{2l+1}{4 \pi} \frac{(l-|m|)!}{(l+|m|)!}}P^{m}_{l}(\cos \theta) e^{im \phi} \times \begin{cases}(-1)^{m} & m\ge 0 \\ 1 & m <0 \end{cases}

##### Explanation

• Spherical Harmonics Equation
• Spherical Harmonics Solution :