Chapter 3
Tables of problems based on basic diﬀerential equation types
3.1 First order ode
3.2 Second order linear ODE
3.3 Second order ode
3.4 Second ODE homogeneous ODE
3.5 Second ODE nonhomogeneous ODE
3.6 Second order nonlinear ODE
3.7 Solved using series method
3.8 Third and higher order ode
3.9 First order ode linear in derivative
3.10 System of diﬀerential equations
3.11 Third and higher order homogeneous ODE
3.12 Third and higher order linear ODE
3.13 Third and higher order nonlinear ODE
3.14 First order ode nonlinear in derivative
3.15 Higher order, nonlinear and homogeneous
3.16 Higher order, nonlinear and nonhomogeneous
3.17 Second order, nonlinear and homogeneous
3.18 Second order, nonlinear and nonhomogeneous
3.19 Third and higher order nonhomogeneous ODE
3.20 Second or higher order ODE with constant coeﬃcients
3.21 Higher order, Linear, Homogeneous and constant coeﬃcients
3.22 Higher order, Linear, Homogeneous and nonconstant coeﬃcients
3.23 Higher order, Linear, nonhomogeneous and constant coeﬃcients
3.24 Second or higher order ODE with nonconstant coeﬃcients
3.25 Second order, Linear, Homogeneous and constant coeﬃcients
3.26 Second order, Linear, Homogeneous and nonconstant coeﬃcients
3.27 Second order, Linear, nonhomogeneous and constant coeﬃcients
3.28 Higher order, Linear, nonhomogeneous and nonconstant coeﬃcients
3.29 Second order, Linear, nonhomogeneous and nonconstant coeﬃcients
This chapter shows how each CAS performed based on the following basic diﬀerential
equations types. A diﬀerential equation is classiﬁed as one of the following types.
 First order ode.
 Second and higher order ode.
For ﬁrst order ode, the following are the main classiﬁcations used.
 First order ode \(f(x,y,y')=0\) which is linear in \(y'(x)\).

First order ode not linear in \(y'(x)\) (such as d’Alembert, Clairaut). But it is important to
note that in this case the ode is nonlinear in \(y'\) when written in the form \(y=g(x,y')\). For an
example, lets look at this ode \[ y' = \frac {x}{2}1+\frac {\sqrt {x^{2}+4 x +4 y}}{2} \] Which is linear in \(y'\) as it stands. But in d’Alembert,
Clairaut we always look at the ode in the form \(y=g(x,y')\). Hence, if we solve for \(y\) ﬁrst, the above
ode now becomes \begin {align*} y &= x y' + \left ( (y')^{2}+ 2 y' + 1 \right )\\ &= g(x,y') \end {align*}
Now we see that \(g(x,y')\) is nonlinear in \(y'\). The above ode happens to be of type
Clairaut.
For second order and higher order ode’s, further classiﬁcation is
 Linear ode.
 nonlinear ode.
Another classiﬁcation for second order and higher order ode’s is
 Constant coeﬃcients ode.
 Varying coeﬃcients ode
Another classiﬁcation for second order and higher order ode’s is
 Homogeneous ode. (the right side is zero).
 Nonhomogeneous ode. (the right side is not zero).
All of the above can be combined to give this classiﬁcation

First order ode.
 First order ode linear in \(y'(x)\).
 First order ode not linear in \(y'(x)\) (such as d’Alembert, Clairaut).

Second and higher order ode

Linear second order ode.
 Linear homogeneous ode. (the right side is zero).
 Linear homogeneous and constant coeﬃcients ode.
 Linear homogeneous and nonconstant coeﬃcients ode.
 Linear nonhomogeneous ode. (the right side is not zero).
 Linear nonhomogeneous and constant coeﬃcients ode.
 Linear nonhomogeneous and nonconstant coeﬃcients ode.

Nonlinear second order ode.
 Nonlinear homogeneous ode.
 Nonlinear nonhomogeneous ode.
For system of diﬀerential equation the following classiﬁcation is used.

System of ﬁrst order odes.
 Linear system of odes.
 nonlinear system of odes.

System of second order odes.
 Linear system of odes.
 nonlinear system of odes.
Each table that follows shows the result per each ODE type.