Notes: `12yy' - 7e^x = 0` Find the general solution of the differential equation

Sunday, 11 December 2016

$\displaystyle{12}{y}{y}'-{7}{{e}}^{{x}}={0}$ Find the general solution of the differential equation

For the given problem: $\displaystyle{12}{y}{y}'-{7}{{e}}^{{x}}={0}$ , we can evaluate this by applying variable separable differential equation in which we express it in a form of $\displaystyle{f{{\left({y}\right)}}}{\left.{d}{y}\right.}={f{{\left({x}\right)}}}{\left.{d}{x}\right.}$ .

Then, $\displaystyle{12}{y}{y}'-{7}{{e}}^{{x}}={0}$ can be rearrange into $\displaystyle{12}{y}{y}'={7}{{e}}^{{x}}$

Express $\displaystyle{y}'$ as $\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}$ :

$\displaystyle{12}{y}\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}={7}{{e}}^{{x}}$

Apply direct integration in the form of $\displaystyle\int{f{{\left({y}\right)}}}{\left.{d}{y}\right.}=\int{f{{\left({x}\right)}}}{\left.{d}{x}\right.}$ :

$\displaystyle{12}{y}\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}={7}{{e}}^{{x}}$

$\displaystyle{12}{y}{\left.{d}{y}\right.}={7}{{e}}^{{x}}{\left.{d}{x}\right.}$

$\displaystyle\int{12}{y}{\left.{d}{y}\right.}=\int{7}{{e}}^{{x}}{\left.{d}{x}\right.}$

For the both side , we apply basic integration property:...

Then, $\displaystyle{12}{y}{y}'-{7}{{e}}^{{x}}={0}$ can be rearrange into $\displaystyle{12}{y}{y}'={7}{{e}}^{{x}}$

Express $\displaystyle{y}'$ as $\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}$ :

$\displaystyle{12}{y}\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}={7}{{e}}^{{x}}$

Apply direct integration in the form of $\displaystyle\int{f{{\left({y}\right)}}}{\left.{d}{y}\right.}=\int{f{{\left({x}\right)}}}{\left.{d}{x}\right.}$ :

$\displaystyle{12}{y}\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}={7}{{e}}^{{x}}$

$\displaystyle{12}{y}{\left.{d}{y}\right.}={7}{{e}}^{{x}}{\left.{d}{x}\right.}$

$\displaystyle\int{12}{y}{\left.{d}{y}\right.}=\int{7}{{e}}^{{x}}{\left.{d}{x}\right.}$

For the both side , we apply basic integration property: $\displaystyle\int{c}\cdot{f{{\left({x}\right)}}}{\left.{d}{x}\right.}={c}\int{f{{\left({x}\right)}}}{\left.{d}{x}\right.}$