A strange 3rd order ODE This is the original ODE: $ y^{1/2}y'''+e^{-x}(y'')^{2+c}-(\frac{xy}{x+1})y'=x $ with c is a positive number. $y(0)=1,y'(0)=0,y''(0)=1$
$1st$ question: If x is large, then $ y^{1/2}y'''$ and $-(\frac{xy}{x+1})y'=x $ are the dominant balance terms,
but what the DB when x is small?
$2nd$ question: Also, I try to solve this ODE with c=0 by using matlab, but matlab keep showing "busy" for over 2hours.. And I really need the solution with c=0.
Can anyone help me? Thanks! 
 A: Answer to question (1):
When $x\to 0, e^{-x}\to 1$, the leading terms in the equation are:
$$y^{1/2}y'''+(y'')^{2+c}=0$$
When $x\to\infty$, $\frac{x}{x+1}\to 1$, so the leading term in the equation are:
$$y^{1/2}y'''-yy'=x$$
A: According to Maple, the first four terms of the series solution of the differential equation around $x=0$ with initial condition 
$y(0)=y_0$, $y'(0) = y_1$, $y''(0)=y_2$, are
$$y \left( x \right) =y_{{0}}+y_{{1}}x+{\frac {y_{{2}}}{2}}{x}^{2}-{
\frac {{{\rm e}^{ \left( 2+c \right) \ln  \left( y_{{2}} \right) }}}{6
}{\frac {1}{\sqrt {y_{{0}}}}}}{x}^{3}+{\frac {1}{48\,y_{{2}}} \left( 2
\,{y_{{0}}}^{3/2} \left( {{\rm e}^{ \left( 2+c \right) \ln  \left( y_{
{2}} \right) }} \right) ^{2}c+4\,{y_{{0}}}^{3/2} \left( {{\rm e}^{
 \left( 2+c \right) \ln  \left( y_{{2}} \right) }} \right) ^{2}+2\,{y_
{{0}}}^{3}y_{{1}}y_{{2}}+2\,{{\rm e}^{ \left( 2+c \right) \ln  \left( 
y_{{2}} \right) }}{y_{{0}}}^{2}y_{{2}}+{{\rm e}^{ \left( 2+c \right) 
\ln  \left( y_{{2}} \right) }}y_{{0}}y_{{1}}y_{{2}}+2\,{y_{{0}}}^{2}y_
{{2}} \right) {y_{{0}}}^{-{\frac {5}{2}}}}{x}^{4}+O \left( {x}^{5}
 \right) 
$$
Of course for this to work you'll want $y_0 > 0$ and $y_2 > 0$.
