JEE Main & Advanced Mathematics Differential Equations Miscellaneous Differential Equation

A special type of second order differential equation : 

\[\frac{{{d}^{2}}y}{d{{x}^{2}}}=f(x)\]                                                                                                                      .....(i)

Equation (i) may be re-written as \[\frac{d}{dx}\left( \frac{dy}{dx} \right)=f(x)\]

\[\Rightarrow \]\[d\left( \frac{dy}{dx} \right)=f(x)dx\]

Integrating, \[\frac{dy}{dx}=\int_{{}}^{{}}{f(x)dx+{{c}_{1}}}\] i.e. \[\frac{dy}{dx}=F(x)+{{c}_{1}}\]                     .....(ii)

Where \[F(x)=\int_{{}}^{{}}{f(x)dx}+{{c}_{1}}dx\]

From (ii), \[dy=f(x)dx+{{c}_{1}}dx\]

Integrating, \[y=\int_{{}}^{{}}{F(x)dx+{{c}_{1}}x+{{c}_{2}}}\]

\[\therefore \]\[y=H(x)+{{c}_{1}}x+{{c}_{2}}\]

where \[H(x)=\int_{{}}^{{}}{F(x)dx}\]  \[{{c}_{1}}\] and \[{{c}_{2}}\] are arbitrary constants.