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Chapter 4
First order linear in derivative

4.1 Flow charts
4.2 ODE form A1 \(y^{\prime }=B+Cf\left ( ax+by+c\right ) \)
4.3 ODE form A2 \(y^{\prime }+p\left ( x\right ) y=q\left ( x\right ) \left ( y\ln y\right ) \)
4.4 Quadrature ode
4.5 Linear ode
4.6 Separable ode
4.7 Homogeneous ode (class A)
4.8 Homogeneous type C \(y^{\prime }=\left ( a+bx+cy\right ) ^{\frac {n}{m}}\)
4.9 Homogeneous Maple type C
4.10 Homogeneous type D
4.11 Homogeneous type D2
4.12 Homogeneous type G
4.13 isobaric ode
4.14 First order special form ID 1 \(y^{\prime }=g\left ( x\right ) e^{a\left ( x\right ) +by}+f\left ( x\right ) \)
4.15 Polynomial ode \(y^{\prime }=\frac {a_{1}x+b_{1}y+c_{1}}{a_{2}x+b_{2}y+c_{2}}\)
4.16 Bernoulli ode \(y^{\prime }+Py=Qy^{n}\)
4.17 Exact ode \(M\left ( x,y\right ) +N\left ( x,y\right ) y^{\prime }=0\)
4.18 Not exact ode but can be made exact with integrating factor
4.19 Not exact first order ode where integrating factor is found by inspection
4.20 Reduced or special Riccati ode \(y^{\prime }=ax^{n}+by^{2}\)
4.21 General Riccati ode \(y^{\prime }=f_{0}+f_{1}y+f_{2}y^{2}\)
4.22 Conversion of Riccati to second order ode
4.23 Abel first order ode
4.24 Chini first order ode \(y^{\prime }=f\left ( x\right ) y^{n}+g\left ( x\right ) y+h\left ( x\right ) \)
4.25 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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