The similarity differential equation $f'''+ff''+\beta f'(f'-1)=0$ with$\beta\textgreater{}0$ is considered. This differential equation appears in thestudy of mixed convection boundary-layer flows over a vertical surface embeddedin a porous medium. In order to prove the existence of solutions satisfying theboundary conditions $f(0)=a\geq0$, $f'(0)=b\geq0$ and $f'(+\infty)=0$ or $1$,we use shooting and consider the initial value problem consisting of thedifferential equation and the initial conditions $f(0)=a$, $f'(0)=b$ and$f''(0)=c$. For $0\textless{}\beta\leq1$, we prove that there exists a unique solutionsuch that $f'(+\infty)=0$, and infinitely many solutions such that$f'(+\infty)=1$. For $\beta\textgreater{}1$, we give only partial results andshow some differences with the previous case.
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