#### Solve

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\centerline{\fbox{\parbox{5.8cm}{\Large\bf UserFunction Solve} }}
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\vspace*{5mm}\noindent{\bf Solve}$(u:ODE1(C,y,x))$
The single argument $u$ has the type $ODE1$ which abbreviates
{\em OrdinaryDifferentialEquation} of order 1, or its subtype $LODE1$. The meaning of the parameters is as follows.
$C:$ Coefficient type.

$y:$ Dependent variable.

$x:$ Independent variable.

The equation must be rational in the dependent variable $y$ and its first derivative.

\vspace*{1mm}\noindent{\bf Specification.}
A Liouvillian fundamental system is returned whenever it exists. In addition, in some cases special function solutions are found.

\vspace*{1mm}\noindent{\bf Examples.}
The input for the examples is given in {\em Reduce} algebraic mode syntax. The output is returned in a separate window.

\4{\tt deq:=Df(y,x)*y**2+1/(3*x)*y**3-2/3=0;}

\4{\tt Solve deq;};

\vspace{1mm}\hspace*{30mm}\fbox{\parbox{11cm}{

The differential equation:
\vspace*{1mm}

$y'y^2+\fracsm{\fracsm{1}{3}}{x}y^3-\fracsm{2}{3}=0$
\vspace{1mm}

The general solution:
\vspace{1mm}

$C+x^2-xy^3=0$  }}

\vspace*{3mm}
\4{\tt deq:=(3*x+2)*(y-2*x-1)*Df(y,x)-y**2+x*y-7*x**2-9*x-3=0;}

\4{\tt Solve deq;}

\vspace{1mm}\hspace*{20mm}\fbox{\parbox{13cm}{

The differential equation:
\vspace{1mm}

${y'}{y}-(2x+1){y'}-\fracsm{\Frac{1}{3}}{x+ \Frac{2}{3}}{y}^{2}+\fracsm{\Frac{1}{3}x}{x+ \Frac{2}{3}} {y}-\fracsm{\Frac{7}{3}x^{2}+3x+1}{x+ \Frac{2}{3}}=0$
\vspace{1mm}

The general solution:
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$Cx+ \Frac{2}{3}C-x^{3}- \Frac{12}{7}x^{2}{y}- \Frac{18}{7}x^{2}- \Frac{3}{7}x{y}^{2}- \Frac{18}{7}x {y}- \Frac{3}{7}x+ \Frac{2}{7}{y}^{3}- \Frac{6}{7}{y}+ \Frac{4}{7}=0$ }}

\vspace*{5mm}\noindent{\bf Solve}$(u:LODE2(RATF Q,y,x))$
The single argument $u$ is a linear ordinary differential equation of order 2 for the unknown function $y$. Its  coefficients are rational functions in the independent variable $x$. Returns Liouvillian solutions or, in some cases, solutions that are equivalent to a special function.

\vspace*{1mm}\noindent{\bf Examples.} The input is given in {\em Reduce} algebraic mode syntax. The output for each minisession is returned in a separate window, $D=d/dx$.

\4{\tt deq:=(x-3)*Df(y,x,2)-(4*x-9)*Df(y,x)+(3*x-6)*y=0;}

\4{\tt s:=Solve deq;}

\vspace{1mm}\hspace*{30mm}\fbox{\parbox{11cm}{

The Differential Equation:

\vspace*{1mm}
${y''}-\fracsm{4x-9}{x-3}{y'}+\fracsm{3x-6}{x-3}{y} = 0$

\vspace*{1mm}
Fundamental system:

\vspace*{1mm}
$y =\{e^{x},\1(\Frac{1}{2}x^{3}- \Frac{21}{4}x^{2} + \Frac{75}{4}x -\Frac{183}{8})e^{3x}\}$ }}

\vspace*{3mm}
\4{\tt deq:=Df(y,x,2)+(x**2-6)/x**2*y=0;}

\4{\tt s:=Solve deq;}

\vspace{1mm}\hspace*{30mm}\fbox{\parbox{11cm}{
Fundamental system:

\vspace{1mm}
$y=\{\fracsm{(x^2+3\theta x-3)e^{\theta x}}{x^2}$

\vspace*{1mm}
Defining polynomial:

\vspace{1mm}
$\theta^2+1=0$ }}

\vspace*{5mm}\noindent{\bf Solve}$(u:LODE3(RATF Q,y,x))$
The single argument $u$ is a linear ordinary differential equation of order 2 for the unknown function $y$. Its  coefficients are rational functions in the independent variable $x$.

\vspace*{1mm}\noindent{\bf Examples.} The input is given in {\em Reduce} algebraic mode syntax. The output for each minisession is returned in a separate window, $D=d/dx$.

\4{\tt deq:=(x**3+x)*Df(y,x,3)+(6*x**2+3)*Df(y,x,2)-12*y=0;}

\4{\tt s:=Solve deq;}

\vspace{1mm}\hspace*{20mm}\fbox{\parbox{13cm}{

The Differential Equation:

${y'''}+\fracsm{6x^{2}+3}{x^{3}+x}{y''}-\fracsm{12}{x^{3}+x}{y} = 0$

\vspace{1mm}

Fundamental system:

\vspace{1mm}
$y =\{x^{2}+ \Frac{1}{2},\1x\sqrt{x^{2}+1}, \1\Frac{3}{2}x\sqrt{x^{2}+1}log{(\sqrt{x^{2}+1}+x-1)}$

\vspace{1mm}\hspace*{20mm}
$-\Frac{3}{2}x\sqrt{x^{2}+1}log{(\sqrt{x^{2}+1}+x+1)}+ \Frac{1}{2}x\}$  }}

\vspace*{5mm}
\4{\tt deq:=Df(y,x,3)+6/x*Df(y,x,2)+(x**2-12)/x**3*y=0;}

\4{\tt s:=Solve deq;}

\vspace{1mm}\hspace*{20mm}\fbox{\parbox{13cm}{

The Differential Equation:

\vspace{1mm}
${y'''}+\fracsm{6}{x}{y''}+\fracsm{x^{3}-12}{x^{3}}{y} = 0$

\vspace*{1mm}
Fundamental system:
\vspace*{1mm}

$y =\{\fracsm{x+2}{x^{3}e^{x}},\1\fracsm{(x+2\theta -2)e^{\theta x}}{x^{3}}\}$
\vspace*{1mm}

Defining polynomial:
\vspace*{1mm}

$\theta ^{2}-\theta +1=0$ }}

\vspace*{5mm}\noindent{\bf Solve}$(u:ODE2(RATF Q,y,x))$
The single argument $u$ is a nonlinear ordinary differential equation of order 2 for the unknown function $y$ that must be rational in the dependent variable $y$ and the independent variable~$x$.

\vspace*{1mm}\noindent{\bf Examples.} The input is given in {\em Reduce} algebraic mode syntax. The output for each minisession is returned in a separate window, $D=d/dx$.

\4{\tt deq:=Df(y,x,2)*y-3*Df(y,x)**2+3*Df(y,x)*y-y**2=0;}

\4{\tt s:=Solve deq;}

\vspace{1mm}\hspace*{20mm}\fbox{\parbox{13cm}{
The Differential Equation:

\vspace{1mm}
${y''}y-3{y'}^{2}+3{y'}y-y^{2}=0$

\vspace{1mm}
The General Solution:

\vspace{1mm}
$C_1y^{2}+e^{2x}C_2+e^{x}=0$  }}

\vspace*{3mm}\noindent{\bf Related Functions:} PolynomialSolution, RationalSolution, Symmetries.

\vspace*{5mm}

{\bf References}

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