Examples of DAEs

This is a simple homogeneous DAE with constant coefficients.
In[1]:=
Click for copyable input
This finds the general solution. It has only one arbitrary constant because the second equation in the system specifies the relationship between x[t] and y[t].
In[2]:=
Click for copyable input
Out[2]=
This verifies the solution.
In[3]:=
Click for copyable input
Out[3]=
Here is an inhomogeneous system derived from the previous example.
In[4]:=
Click for copyable input
The general solution is composed of the general solution to the corresponding homogeneous system and a particular solution to the inhomogeneous equation.
In[5]:=
Click for copyable input
Out[5]=
This solves an initial value problem for the previous equation.
In[6]:=
Click for copyable input
In[7]:=
Click for copyable input
Out[7]=
Here is a plot of the solution and the constraint (algebraic) condition.
In[8]:=
Click for copyable input
Out[8]=
In this DAE, the inhomogeneous part is quite general.
In[9]:=
Click for copyable input
In[10]:=
Click for copyable input
Note that there are no degrees of freedom in the solution (that is, there are no arbitrary constants) because z[t] is given algebraically, and thus x[t] and y[t] can be determined uniquely from z[t] using differentiation.
In[11]:=
Click for copyable input
Out[11]=
In[12]:=
Click for copyable input
Out[12]=
In this example, the algebraic constraint is present only implicitly: all three equations contain derivatives of the unknown functions.
In[13]:=
Click for copyable input
In[14]:=
Click for copyable input
The Jacobian with respect to the derivatives of the unknown functions is singular, so that it is not possible to solve for them.
In[15]:=
Click for copyable input
Out[15]=
In[16]:=
Click for copyable input
Out[16]=
The differential-algebraic character of this problem is clear from the smaller number of arbitrary constants (two rather than three) in the general solution.
In[17]:=
Click for copyable input
Out[17]=

Systems of equations with higher-order derivatives are solved by reducing them to first-order systems.

Here is the general solution to a homogeneous DAE of order two with constant coefficients.
In[18]:=
Click for copyable input
In[19]:=
Click for copyable input
Out[19]=
In[20]:=
Click for copyable input
Out[20]=
This inhomogeneous system of ODEs is based on the previous example.
In[21]:=
Click for copyable input
In[22]:=
Click for copyable input
Out[22]=
In[23]:=
Click for copyable input
Out[23]=
Here is an initial value problem for the previous system of equations.
In[24]:=
Click for copyable input
In[25]:=
Click for copyable input
Out[25]=
Here is a plot of the solution.
In[26]:=
Click for copyable input
Out[26]=
Finally, here is a system with a third-order ODE. Since the coefficients are exact quantities, the computation takes some time.
In[27]:=
Click for copyable input
In[28]:=
Click for copyable input
In[29]:=
Click for copyable input
Out[29]=
In[30]:=
Click for copyable input
Out[30]=

The symbolic solution of DAEs that are nonlinear or have non-constant coefficients is a difficult problem. Such systems can often be solved numerically with the Wolfram Language function NDSolve.

Translate this page: