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Differential Equations

fricas
y := operator y

\label{eq1}y(1)
fricas
deq := D(y(x),x)=(1+y(x)^2)/(1+2*x*y(x))

\label{eq2}{{y_{\ }^{,}}\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}{{{y_{\ }^{, , ,}}\left({x}\right)}+{{y_{\ }^{, ,}}\left({x}\right)}+{{y_{\ }^{,}}\left({x}\right)}+{y \left({x}\right)}}={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_{\ }^{,}}\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_{\ }^{,}}\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)
fricas
deq2 := D(z(x),x) = exp(y(x))/(1 - d * y(x))

\label{eq10}{{z_{\ }^{,}}\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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