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Chapter 2
First order linear in derivative

2.1 Flow charts
2.2 ODE form A1 \(y^{\prime }=B+Cf\left ( ax+by+c\right ) \)
2.3 ODE form A2 \(y^{\prime }+p\left ( x\right ) y=q\left ( x\right ) \left ( y\ln y\right ) \)
2.4 Quadrature ode
2.5 Linear ode
2.6 Separable ode
2.7 Homogeneous ode (class A)
2.8 Homogeneous type C \(y^{\prime }=\left ( a+bx+cy\right ) ^{\frac {n}{m}}\)
2.9 Homogeneous Maple type C
2.10 Homogeneous type D
2.11 Homogeneous type D2
2.12 Homogeneous type G
2.13 isobaric ode
2.14 First order special form ID 1 \(y^{\prime }=g\left ( x\right ) e^{a\left ( x\right ) +by}+f\left ( x\right ) \)
2.15 Polynomial ode \(y^{\prime }=\frac {a_{1}x+b_{1}y+c_{1}}{a_{2}x+b_{2}y+c_{2}}\)
2.16 Bernoulli ode \(y^{\prime }+Py=Qy^{n}\)
2.17 Exact ode \(M\left ( x,y\right ) +N\left ( x,y\right ) y^{\prime }=0\)
2.18 Not exact ode but can be made exact with integrating factor
2.19 Not exact first order ode where integrating factor is found by inspection
2.20 General Riccati ode \(y^{\prime }=f_{0}+f_{1}y+f_{2}y^{2}\)
2.21 Abel ode \(y^{\prime }=f_{0}+f_{1}y+f_{2}y^{2}+f_{3}y^{3}\)
2.22 Chini first order ode \(y^{\prime }=f\left ( x\right ) y^{n}+g\left ( x\right ) y+h\left ( x\right ) \)
2.23 differential type ode \(y^{\prime }=f\left ( x,y\right ) \)

These are first order ode’s of form \(F\left ( x,y,y^{\prime }\right ) =0\) which are linear in \(y^{\prime }\) but can be nonlinear in \(y\).

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