TITLE: Is there an Analytical solution for Blasius equation?
QUESTION [5 upvotes]: Blasius Equation was introduced to me during my University time ...and I would like to have a solution for it 
$$y''' + yy'' = 0$$
the $y$ here makes the equation from a linear simple to solve to a non-linear almost impossible to solve , one solution is to use the numerical method but I would like to have the exact analytic solution .

REPLY [5 votes]: Of course, there's one simple two-parameter family of solutions: $y(x) = a + b x$.
You might look at series solutions:
$$\eqalign{y \left( x \right) &=a+bx+c{x}^{2}-{\frac {ac}{3}}{x}^{3}+ \left( {
\frac {{a}^{2}c}{12}}-{\frac {bc}{12}} \right) {x}^{4}+ \left( -{
\frac {{a}^{3}c}{60}}+{\frac {bac}{20}}-{\frac {{c}^{2}}{30}} \right) 
{x}^{5}\cr &+ \left( {\frac {{c}^{2}a}{36}}+{\frac {{a}^{4}c}{360}}-{\frac 
{b{a}^{2}c}{60}}+{\frac {{b}^{2}c}{120}} \right) {x}^{6}+ \left( -{
\frac {4\,{a}^{2}{c}^{2}}{315}}+{\frac {11\,b{c}^{2}}{1260}}-{\frac {{
a}^{5}c}{2520}}+{\frac {b{a}^{3}c}{252}}-{\frac {{b}^{2}ac}{168}}
 \right) {x}^{7}\cr &+ \left( {\frac {{a}^{3}{c}^{2}}{240}}-{\frac {ba{c}^{
2}}{120}}+{\frac {{a}^{6}c}{20160}}-{\frac {b{a}^{4}c}{1344}}+{\frac {
{b}^{2}{a}^{2}c}{448}}-{\frac {{b}^{3}c}{1344}}+{\frac {11\,{c}^{3}}{
5040}} \right) {x}^{8}\cr &+ \left( -{\frac {13\,{c}^{3}a}{5040}}-{\frac {
11\,{a}^{4}{c}^{2}}{10080}}+{\frac {4\,b{a}^{2}{c}^{2}}{945}}-{\frac {
43\,{b}^{2}{c}^{2}}{30240}}-{\frac {{a}^{7}c}{181440}}+{\frac {b{a}^{5
}c}{8640}}-{\frac {{b}^{2}{a}^{3}c}{1728}}+{\frac {{b}^{3}ac}{1728}}
 \right) {x}^{9}\cr & + \ldots}
$$
In particular the case $a=b=0$ is rather nice:
$$ \eqalign{y \left( x \right) &=c{x}^{2}-{\frac {{c}^{2}}{30}}{x}^{5}+{\frac {11
\,{c}^{3}}{5040}}{x}^{8}-{\frac {5\,{c}^{4}}{33264}}{x}^{11}+{\frac {
9299\,{c}^{5}}{908107200}}{x}^{14}-{\frac {1272379\,{c}^{6}}{
1852538688000}}{x}^{17}\cr& +{\frac {19241647\,{c}^{7}}{422378820864000}}{x
}^{20}-{\frac {33675995567\,{c}^{8}}{11220493376252160000}}{x}^{23}\cr &+{
\frac {17241364408921\,{c}^{9}}{87519848334766848000000}}{x}^{26}-{
\frac {1375703592341009\,{c}^{10}}{106599175271746020864000000}}{x}^{
29}+\ldots}
$$
EDIT: The coefficient of $x^{3n-1}$ here is $- a(n-1) (-c)^n/(3n-1)!$ where $a(n)$ is OEIS sequence A018893.
EDIT: See Bluman and Anco, "Symmetry and Integration Methods for Differential Equations", sec. 3.4.2, where the Blasius equation is reduced to a first order equation.