On the asymptotic solutions of a class of ordinary differential equations of the fourth order. I. Existence of regular formal solutions

Authors:
C. C. Lin and A. L. Rabenstein

Journal:
Trans. Amer. Math. Soc. **94** (1960), 24-57

MSC:
Primary 34.00; Secondary 76.00

DOI:
https://doi.org/10.1090/S0002-9947-1960-0110865-8

MathSciNet review:
0110865

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References | Similar Articles | Additional Information

**[1]**Rudolph E. Langer,*The asymptotic solutions of ordinary linear differential equations of the second order, with special reference to a turning point*, Trans. Amer. Math. Soc.**67**(1949), 461–490. MR**0033420**, https://doi.org/10.1090/S0002-9947-1949-0033420-2**[2]**Rudolph E. Langer,*On the asymptotic forms of the solutions of ordinary linear differential equations of the third order in a region containing a turning point*, Trans. Amer. Math. Soc.**80**(1955), 93–123. MR**0073009**, https://doi.org/10.1090/S0002-9947-1955-0073009-5**[3]**-,*The solutions of the differential equation*, Duke Math. J. vol. 22 (1955b) pp. 525-542.**[4]**Rudolph E. Langer,*On the asymptotic solutions of a class of ordinary differential equations of the fourth order, with special reference to an equation of hydrodynamics*, Trans. Amer. Math. Soc.**84**(1957), 144–191. MR**0083637**, https://doi.org/10.1090/S0002-9947-1957-0083637-0**[5]**Rudolph E. Langer,*Formal solutions and a related equation for a class of fourth order differential equations of a hydrodynamic type*, Trans. Amer. Math. Soc.**92**(1959), 371–410. MR**0109923**, https://doi.org/10.1090/S0002-9947-1959-0109923-5**[6]**C. C. Lin,*The theory of hydrodynamic stability*, Cambridge, at the University Press, 1955. MR**0077331****[7]**-,*On uniformly valid asymptotic solutions of the Orr-Sommerfeld equation*, Proceedings of the Ninth International Congress of Applied Mechanics (Brussels) vol. I (1957) pp. 136-148.**[8]**-,*On the stability of the laminar boundary layer*, Symposium on Naval Hydro dynamics (National Research Council Publication 515), Washington, D. C., 1958, pp. 353-371.**[9]**-,*On the instability of laminar flow and its transition to turbulence*, Proceedings of the Symposium on Boundary Layer Theory (Freiburg), Springer, 1958, pp. 144-160.**[10]**Albert L. Rabenstein,*Asymptotic solutions of 𝑢^{𝑖𝑣}+𝜆²(𝑧𝑢”+𝛼𝑢’+𝛽𝑢)=0 for large |𝜆|*, Arch. Rational Mech. Anal.**1**(1958), 418–435. MR**0096855**, https://doi.org/10.1007/BF00298019**[11]**Wolfgang Wasow,*A study of the solutions of the differential equation 𝑦⁽⁴⁾+𝜆²(𝑥𝑦”+𝑦)=0 for large values of 𝜆*, Ann. of Math. (2)**52**(1950), 350–361. MR**0037432**, https://doi.org/10.2307/1969474**[12]**Wolfgang Wasow,*Asymptotic solution of the differential equation of hydrodynamic stability in a domain containing a transition point*, Ann. of Math. (2)**58**(1953), 222–252. MR**0061737**, https://doi.org/10.2307/1969787**[13]**-, Discussion following paper by Lin, [8], ibid., pp. 363-366.

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DOI:
https://doi.org/10.1090/S0002-9947-1960-0110865-8

Article copyright:
© Copyright 1960
American Mathematical Society