Substituting the values of the initial conditions will give particular solutions.

### Examples:

Here is the good demonstration of Differential equations and Differentiation.

Rather than solving it manually, we can solve it using Sage.

## Sage (System for Algebra and Geometry Experimentation) :

Sage is mathematical software with features covering many aspects of mathematics, including algebra, combinatorics, numerical mathematics, number theory, and calculus.

Ways to use Sage:- Notebook Graphical Interface
- Interactive Command Line
- Programs
- Scripts

This link will help you to solve DE in Sage.

## >>>We can Embed Sage with Tex using SageTex:<<<

The SageTeX package allows you to embed the results of Sage computations into a LaTeX document. It comes standard with Sage.

For its Installation, click here.

Example of SageTex file:

Using Sage\TeX, one can use Sage to compute things and put them into

your \LaTeX{} document.

Below is the example of Sage code with LateX:

\documentclass{article} \usepackage{sagetex} \begin{document} Using Sage\TeX, one can use Sage to compute things and put them into your \LaTeX{} document. Here's some Sage code: \begin{sageblock} f(x) = exp(x) * sin(2*x) \end{sageblock} The second derivative of $f$ is \[ \frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}} \sage{f(x)} = \sage{diff(f, x, 2)(x)}. \] Here's a plot of $f$ from $-1$ to $1$: \sageplot{plot(f, -1, 1)} \end{document}

A sageblock environment typesets your code verbatim and also executes the code when you run Sage. \sage{foo}, the result put into your document is whatever you get from running latex(foo) inside Sage. /sageplot is used to plot a graph.

How to run SageTeX code:- Run LaTeX on your .tex file
- Run Sage on the generated .sage file
- Run LaTeX again.

To read the documentation for SageTeX,

`$ cd SAGE_ROOT/local/share/texmf/tex/generic/sagetex`

SAGE_ROOT is the path where you have installed Sage. In my system, SAGE_ROOT = /home/jasleen/sage/