An exact solution of the nonlinear differential equation 𝑦̈+𝑝(𝑡)𝑦=𝑞ₘ(𝑡)/𝑦^{2𝑚-1}

Reid, James L.

doi:10.1090/S0002-9939-1971-0269907-4

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

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An exact solution of the nonlinear differential equation $\ddot y+p(t)y=q_{m} (t)/y^{2m-1}$
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by James L. Reid

Proc. Amer. Math. Soc. 27 (1971), 61-62

DOI: https://doi.org/10.1090/S0002-9939-1971-0269907-4

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Abstract:

An exact solution of the nonlinear differential equation $\ddot y + p(t)y = {q_m}(t)/{y^{2m - 1}}$ is found to be $y = {[{u^m} + c{(m - 1)^{ - 1}}{W^{ - 2}}v]^{1/m}}$ if ${q_m}(t) = c{(uv)^{m - 2}}$. $u$ and $v$ are independent solutions of $\ddot y + p(t)y = 0$ and $W$ is their Wronskian.

References

Edmund Pinney, The nonlinear differential equation $y''+p(x)y+cy^{-3}=0$, Proc. Amer. Math. Soc. 1 (1950), 681. MR 37979, DOI 10.1090/S0002-9939-1950-0037979-4