FCDiffEqSolve
FCDiffEqSolve[mat, var, eps, n]
constructs a solution for a single-variable differential equation G ′ = ε B G G' = \varepsilon \mathcal{B} G G ′ = ε B G in the canonical form, where mat
is B B B , var
is the variable w.r.t. which G G G was differentiated and n
is the required order in eps
.
The output consists of iterated integrals written in terms of FCIteratedIntegral
objects.
See also
Overview , FCIteratedIntegral , FCDiffEqChangeVariables
Examples
mat = {{ - 2 / x , 0 , 0 }, { 0 , 0 , 0 }, { - x ^ (- 1 ), 3 / x , - 2 / x }}
( − 2 x 0 0 0 0 0 − 1 x 3 x − 2 x ) \left(
\begin{array}{ccc}
-\frac{2}{x} & 0 & 0 \\
0 & 0 & 0 \\
-\frac{1}{x} & \frac{3}{x} & -\frac{2}{x} \\
\end{array}
\right) − x 2 0 − x 1 0 0 x 3 0 0 − x 2
FCDiffEqSolve[ mat, x , ep, 1 ]
{ ep ( C [ 1 , 0 ] FCIteratedIntegral ( FCPartialFractionForm ( 0 , ( { x , − 1 } − 2 ) , x ) , x , 0 , x ) + C [ 1 , 1 ] ) + C [ 1 , 0 ] , ep C [ 2 , 1 ] + C [ 2 , 0 ] , ep ( C [ 3 , 0 ] FCIteratedIntegral ( FCPartialFractionForm ( 0 , ( { x , − 1 } − 2 ) , x ) , x , 0 , x ) + C [ 1 , 0 ] FCIteratedIntegral ( FCPartialFractionForm ( 0 , ( { x , − 1 } − 1 ) , x ) , x , 0 , x ) + C [ 2 , 0 ] FCIteratedIntegral ( FCPartialFractionForm ( 0 , ( { x , − 1 } 3 ) , x ) , x , 0 , x ) + C [ 3 , 1 ] ) + C [ 3 , 0 ] } \left\{\text{ep} \left(C[1,0] \;\text{FCIteratedIntegral}\left(\text{FCPartialFractionForm}\left(0,\left(
\begin{array}{cc}
\{x,-1\} & -2 \\
\end{array}
\right),x\right),x,0,x\right)+C[1,1]\right)+C[1,0],\text{ep} C[2,1]+C[2,0],\text{ep} \left(C[3,0] \;\text{FCIteratedIntegral}\left(\text{FCPartialFractionForm}\left(0,\left(
\begin{array}{cc}
\{x,-1\} & -2 \\
\end{array}
\right),x\right),x,0,x\right)+C[1,0] \;\text{FCIteratedIntegral}\left(\text{FCPartialFractionForm}\left(0,\left(
\begin{array}{cc}
\{x,-1\} & -1 \\
\end{array}
\right),x\right),x,0,x\right)+C[2,0] \;\text{FCIteratedIntegral}\left(\text{FCPartialFractionForm}\left(0,\left(
\begin{array}{cc}
\{x,-1\} & 3 \\
\end{array}
\right),x\right),x,0,x\right)+C[3,1]\right)+C[3,0]\right\} { ep ( C [ 1 , 0 ] FCIteratedIntegral ( FCPartialFractionForm ( 0 , ( { x , − 1 } − 2 ) , x ) , x , 0 , x ) + C [ 1 , 1 ] ) + C [ 1 , 0 ] , ep C [ 2 , 1 ] + C [ 2 , 0 ] , ep ( C [ 3 , 0 ] FCIteratedIntegral ( FCPartialFractionForm ( 0 , ( { x , − 1 } − 2 ) , x ) , x , 0 , x ) + C [ 1 , 0 ] FCIteratedIntegral ( FCPartialFractionForm ( 0 , ( { x , − 1 } − 1 ) , x ) , x , 0 , x ) + C [ 2 , 0 ] FCIteratedIntegral ( FCPartialFractionForm ( 0 , ( { x , − 1 } 3 ) , x ) , x , 0 , x ) + C [ 3 , 1 ] ) + C [ 3 , 0 ] }