List of Differential Equations Formulas Latex Code
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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
- 1.1 Wave Equation
- 1.2 Heat Equation
- 1.3 Laplace Equation
- 1.4 Legendre's Equation
- 1.5 Bessel's Equation
- 1.6 Spherical Harmonics Equation
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1.0 Diffusion (conduction) Equation
Equation
Latex Code
\kappa u_{xx} = u_{t} \\ \frac{\partial \Psi}{\partial t}=\kappa \triangledown^{2} \PsiExplanation
: First Order Partial Derivative of function u(x,t) over time t.
: Second Order Partial Derivative of function u(x,t) over x.
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1.1 Wave Equation
Equation
Latex Code
u_{tt}=c^{2}u_{xx}Explanation
: Second Order Partial Derivative of function u(x,t) over time t.
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1.2 Heat Equation
Equation
Latex Code
u_{t}={\alpha}^{2}u_{xx}Explanation
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1.3 Laplace Equation
Equation
Latex Code
u_{xx} = 0Explanation
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1.4 Legendre's Equation
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:
Related Documents
- Solutions of Legendre equations are Legendre polynomials
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1.5 Bessel's equation
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
.
Related Documents
- Solutions of Bessel equations
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1.6 Spherical Harmonics Equation
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
:
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1. Differential Equations
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