Miscellaneous Differential Equation
Category : JEE Main & Advanced
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.
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