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3.3 First order linear in derivative

3.3.1 Flow charts
3.3.2 ODE of form \(y^{\prime }=B+Cf\left ( ax+by+c\right ) \)
3.3.3 ODE of form \(y^{\prime }+p\left ( x\right ) y=q\left ( x\right ) \left ( y\ln y\right ) \)
3.3.4 Quadrature ode
3.3.5 Linear ode
3.3.6 Separable ode
3.3.7 Homogeneous ode (class A)
3.3.8 Homogeneous type C \(y^{\prime }=\left ( a+bx+cy\right ) ^{\frac {n}{m}}\)
3.3.9 Homogeneous Maple type C
3.3.10 Homogeneous type D
3.3.11 Homogeneous type D2
3.3.12 Homogeneous type G
3.3.13 isobaric ode
3.3.14 First order special form ID 1 \(y^{\prime }=g\left ( x\right ) e^{a\left ( x\right ) +by}+f\left ( x\right ) \)
3.3.15 Polynomial ode \(y^{\prime }=\frac {a_{1}x+b_{1}y+c_{1}}{a_{2}x+b_{2}y+c_{2}}\)
3.3.16 Bernoulli ode \(y^{\prime }+Py=Qy^{n}\)
3.3.17 Exact ode \(M\left ( x,y\right ) +N\left ( x,y\right ) y^{\prime }=0\)
3.3.18 Not exact ode but can be made exact with integrating factor
3.3.19 Not exact first order ode where integrating factor is found by inspection
3.3.20 Riccati ode \(y^{\prime }=f_{0}+f_{1}y+f_{2}y^{2}\)
3.3.21 Abel first kind ode \(y^{\prime }=f_{0}+f_{1}y+f_{2}y^{2}+f_{3}y^{3}\)
3.3.22 Chini first order ode \(y^{\prime }=f\left ( x\right ) \left ( y^{\prime }\right ) ^{n}+g\left ( x\right ) y+h\left ( x\right ) \)
3.3.23 differential type ode \(y^{\prime }=f\left ( x,y\right ) \)
3.3.24 Series method
3.3.25 Laplace method
\[ F\left ( x,y,y^{\prime }\right ) =0 \]

These are first order ode’s which are linear in \(y^{\prime }\) but can be nonlinear in \(y\).

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