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1.2 First order linear in derivative

1.2.1 Flow charts
1.2.2 Quadrature ode
1.2.3 Linear ode
1.2.4 Separable ode
1.2.5 Homogeneous ode
1.2.6 Homogeneous type C \(y^{\prime }=\left ( a+bx+cy\right ) ^{\frac {n}{m}}\)
1.2.7 Homogeneous Maple type C
1.2.8 Homogeneous type D
1.2.9 Homogeneous type D2
1.2.10 isobaric ode
1.2.11 First order special form ID 1 \(y^{\prime }=g\left ( x\right ) e^{a\left ( x\right ) +by}+f\left ( x\right ) \)
1.2.12 Polynomial ode \(y^{\prime }=\frac {a_{1}x+b_{1}y+c_{1}}{a_{2}x+b_{2}y+c_{2}}\)
1.2.13 Bernoulli ode \(y^{\prime }+Py=Qy^{n}\)
1.2.14 Exact ode \(M\left ( x,y\right ) +N\left ( x,y\right ) y^{\prime }=0\)
1.2.15 Not exact ode but can be made exact with integrating factor
1.2.16 Not exact first order ode where integrating factor is found by inspection
1.2.17 Riccati ode \(y^{\prime }=f_{0}+f_{1}y+f_{2}y^{2}\)
1.2.18 Abel first kind ode \(y^{\prime }=f_{0}+f_{1}y+f_{2}y^{2}+f_{3}y^{3}\)
1.2.19 differential type ode \(y^{\prime }=f\left ( x,y\right ) \)
1.2.20 Series method
1.2.21 Laplace method
1.2.22 Lie symmetry method for solving first order ODE

\[ F\left ( x,y,y^{\prime }\right ) =0 \]

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

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