MathAction SandBoxDifferentialEquations

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Editor: Bill Page Time: 2009/10/16 19:44:19 GMT-7
Note: merge

changed:
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Differential Equations

\begin{axiom}
y := operator y
deq := D(y(x),x)=(1+y(x)^2)/(1+2*x*y(x))
solve(deq,y,x)
deq := D(y(x),x,3)+D(y(x),x,2)+D(y(x),x,1)+y(x)=x*sin(x)
solve(deq,y,x)
\end{axiom}

test solution

\begin{axiom}
deq := D(y(x),x)=(a+y(x)^2)/(b+2*x^2*y(2*x))
solve(deq,y,x)
deq := D(y(x),x,3)+D(y(x),x,2)+D(y(x),x,1)+y(x)=x*sin(x)
solve(deq,y,x)
\end{axiom}

Linear non homog equation
\begin{axiom}
deq := D(y(x),x) = y(x)^2 + 2 * y(x) + 1
solve(deq,y,x)
z := operator z
deq2 := D(z(x),x) = exp(y(x))/(1 - d * y(x))
solve(deq2,z,x)
\end{axiom}

Differential Equations

fricas

y := operator y

$\label{eq1}y$

(1)

Type: BasicOperator?

fricas

deq := D(y(x),x)=(1+y(x)^2)/(1+2*x*y(x))

$\label{eq2}{{y^{\prime}}\left({x}\right)}={{{{y \left({x}\right)}^{2}}+ 1}\over{{2 \ x \ {y \left({x}\right)}}+ 1}}$

(2)

Type: Equation(Expression(Integer))

fricas

solve(deq,y,x)

$\label{eq3}{{{\left({{y \left({x}\right)}^{2}}+ 1 \right)}\ {\arctan \left({y \left({x}\right)}\right)}}+{y \left({x}\right)}-{2 \ x}}\over{{2 \ {{y \left({x}\right)}^{2}}}+ 2}$

(3)

Type: Union(Expression(Integer),...)

fricas

deq := D(y(x),x,3)+D(y(x),x,2)+D(y(x),x,1)+y(x)=x*sin(x)

$\label{eq4}\begin{array}{@{}l} \displaystyle {{{y^{\prime \prime \prime}}\left({x}\right)}+{{y^{\prime \prime}}\left({x}\right)}+{{y^{\prime}}\left({x}\right)}+{y \left({x}\right)}}= \ \ \displaystyle {x \ {\sin \left({x}\right)}}$

(4)

Type: Equation(Expression(Integer))

fricas

solve(deq,y,x)

$\label{eq5}\begin{array}{@{}l} \displaystyle \left[{ \begin{array}{@{}l} \displaystyle particular ={{\left( \begin{array}{@{}l} \displaystyle {{\left({2 \ x}+ 1 \right)}\ {\cos \left({x}\right)}\ {{\sin \left({x}\right)}^{2}}}+ \ \ \displaystyle {{\left(-{{x}^{2}}+{3 \ x}\right)}\ {\sin \left({x}\right)}}+{{\left({2 \ x}+ 1 \right)}\ {{\cos \left({x}\right)}^{3}}}+ \ \ \displaystyle {{\left(-{{x}^{2}}-{3 \ x}+ 2 \right)}\ {\cos \left({x}\right)}}$

(5)

Type: Union(Record(particular: Expression(Integer),basis: List(Expression(Integer))),...)

test solution

fricas

deq := D(y(x),x)=(a+y(x)^2)/(b+2*x^2*y(2*x))

$\label{eq6}{{y^{\prime}}\left({x}\right)}={{{{y \left({x}\right)}^{2}}+ a}\over{{2 \ {{x}^{2}}\ {y \left({2 \ x}\right)}}+ b}}$

(6)

Type: Equation(Expression(Integer))

fricas

solve(deq,y,x)

   >> Error detected within library code:
   getfreelincoeff: not a linear ordinary differential equation

Linear non homog equation

fricas

deq := D(y(x),x) = y(x)^2 + 2 * y(x) + 1

$\label{eq7}{{y^{\prime}}\left({x}\right)}={{{y \left({x}\right)}^{2}}+{2 \ {y \left({x}\right)}}+ 1}$

(7)

Type: Equation(Expression(Integer))

fricas

solve(deq,y,x)

$\label{eq8}{-{x \ {y \left({x}\right)}}- x - 1}\over{{y \left({x}\right)}+ 1}$

(8)

Type: Union(Expression(Integer),...)

fricas

z := operator z

$\label{eq9}z$

(9)

Type: BasicOperator?

fricas

deq2 := D(z(x),x) = exp(y(x))/(1 - d * y(x))

$\label{eq10}{{z^{\prime}}\left({x}\right)}= -{{{e}^{y \left({x}\right)}}\over{{d \ {y \left({x}\right)}}- 1}}$

(10)

Type: Equation(Expression(Integer))

fricas

solve(deq2,z,x)

$\label{eq11}\left[{particular ={\int^{ \displaystyle x}-{{{{e}^{y \left({\%D}\right)}}\over{{d \ {y \left({\%D}\right)}}- 1}}\ {d \%D}}}}, \:{basis ={\left[ 1 \right]}}\right]$

(11)

Type: Union(Record(particular: Expression(Integer),basis: List(Expression(Integer))),...)

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