Linear ordinary differential equations of first order

 

Differential equations are equation which are made of  derivative of dependent variable, say $\frac{dy}{dx}$, independent variable, say $\mathcal x$ & dependent variable, say $\mathcal y$.

Mathematically they are written as  $f(x,y,\frac{dy}{dx})=0$

As you know there are many problems in real world which are solved using differential equations.  

Linear equations

$\frac{dy}{dx}+Py=Q$

$I.F.=e^\int Pdx$ &

 $y.e^\int Pdx=\int Qe^\int Pdx +c$

$y.IF=\int Q.IF +c$

 or

 $\frac{dx}{dy}+Px=Q$

$I.F.=e^\int Pdy$ & 

$y.e^\int Pdy=\int Qe^\int Pdy +c$

$x.IF=\int Q.IF +c$

Also know Trajectories