# Solve Differential Equation dy/dx=y/x

In this tutorial we shall evaluate the simple differential equation of the form $\frac{{dy}}{{dx}} = \frac{y}{x}$, and we shall use the method of separating the variables.

The differential equation of the form is given as

$\frac{{dy}}{{dx}} = \frac{y}{x}$

Separating the variables, the given differential equation can be written as

$\frac{1}{y}dy = \frac{1}{x}dx\,\,\,\,{\text{ – – – }}\left( {\text{i}} \right)$

With the separating the variable technique we must keep the terms $dy$ and $dx$ in the numerators with their respective functions.

Now integrating both sides of the equation (i), we have
$\int {\frac{1}{y}dy = \int {\frac{1}{x}dx} }$

Using the formula of integration $\int {\frac{1}{x}dx = \ln x + c}$, we get
$\begin{gathered} \ln y = \ln x + \ln c \\ \Rightarrow \ln y = \ln xc \\ \end{gathered}$

Cancelling the logarithm from both sides of the above equation, we get
$y = xc$

This is the required solution of the given differential equation.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9939448237419128, "perplexity": 150.2730425511997}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496668782.15/warc/CC-MAIN-20191117014405-20191117042405-00387.warc.gz"}