Notes: `yy' = 4sinx` Find the general solution of the differential equation

Wednesday, 29 June 2016

$\displaystyle{y}{y}'={4}{\sin{{x}}}$ Find the general solution of the differential equation

The general solution of a differential equation in a form of $\displaystyle{y}'={f{{\left({x}\right)}}}$ can

be evaluated using direct integration. The derivative of y denoted as $\displaystyle{y}'$ can be written as $\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}$ then $\displaystyle{y}'={f{{\left({x}\right)}}}$ can be expressed as $\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}={f{{\left({x}\right)}}}$ .

For the problem $\displaystyle{y}{y}'={4}{\sin{{\left({x}\right)}}}$ , we may apply $\displaystyle{y}'=\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}$ to set-up the integration:

$\displaystyle{y}\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}={4}{\sin{{\left({x}\right)}}}$ .

or $\displaystyle{y}{\left.{d}{y}\right.}={4}{\sin{{\left({x}\right)}}}{\left.{d}{x}\right.}$

Then set-up direct integration on...

The general solution of a differential equation in a form of $\displaystyle{y}'={f{{\left({x}\right)}}}$ can

For the problem $\displaystyle{y}{y}'={4}{\sin{{\left({x}\right)}}}$ , we may apply $\displaystyle{y}'=\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}$ to set-up the integration:

$\displaystyle{y}\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}={4}{\sin{{\left({x}\right)}}}$ .

or $\displaystyle{y}{\left.{d}{y}\right.}={4}{\sin{{\left({x}\right)}}}{\left.{d}{x}\right.}$

Then set-up direct integration on both sides:

$\displaystyle\int{y}{\left.{d}{y}\right.}=\int{4}{\sin{{\left({x}\right)}}}{\left.{d}{x}\right.}$

Integration:

Apply Power Rule integration: $\displaystyle\int{{u}}^{{n}}{d}{u}=\frac{{{u}}^{{{n}+{1}}}}{{{n}+{1}}}$ on $\displaystyle\int{y}{\left.{d}{y}\right.}$ .

Note: $\displaystyle{y}$ is the same as $\displaystyle{{y}}^{{1}}$ .

$\displaystyle\int{y}{\left.{d}{y}\right.}=\frac{{{y}}^{{{1}+{1}}}}{{{1}+{1}}}$

$\displaystyle=\frac{{{y}}^{{2}}}{{2}}$

Apply the basic integration property: $\displaystyle\int{c}\cdot{f{{\left({x}\right)}}}{\left.{d}{x}\right.}={c}\int{f{{\left({x}\right)}}}{\left.{d}{x}\right.}$ and basic integration formula for sine function: $\displaystyle\int{\sin{{\left({u}\right)}}}{d}{u}=-{\cos{{\left({u}\right)}}}+{C}$