Linear first order ODEs
A first order ODE is said to be linear if it can be brought into the form
$$ y’+P(x)y=Q(x) $$
It’s called “homogeneous linear first order ODE” when Q(x)=0
Solution #
$$ \begin{align*} P(x)ydx+dy=Q(x)dx\\ (Py-Q)dx+dy=0\\ I.F. = e^{\int Pdx}=e^h\\ \\ Solution: \int M + \int (terms\\ of\\ N\\ not\\ containing\\ x)\\ \int e^{\int Pdx}[Py-Q]dx=0\\ \int e^{\int Pdx}Py\\ dx=\int e^hQ\\ dx\\ Since,\\ d(e^{h})=Pe^{\int P dx}=Pe^{h}dx\\ \int yd(e^h)=\int hQ dx\\\ \
ye^h=\int e^h Q dx +c\\ or\\ y(x)=e^{-h}\left(\int e^{h}Q dx +c\right)\\ I.F.y=\int I.F.Qdx +c \end{align*} $$ We use Exact ODE and Integrating Factors#If I.F. is a function of x in the third line
Bernoulli’s Equation #
Nonlinear ODE → Linear ODE $$ \begin{align*} y’+P(x)\\ y&= Q(x)y^a\\ transformed\\ to\\ \\ u’+(1-a)P\\ u &= (1-a)Q(x)\\ [u=y^{1-a}] \end{align*} $$