83.29.4 problem 4

Internal problem ID [19377]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VII. Exact differential equations and certain particular forms of equations. Exercise VII (C) at page 107
Problem number : 4
Date solved : Tuesday, January 28, 2025 at 01:35:51 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} y^{\prime \prime }+\frac {a^{2}}{y}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.026 (sec). Leaf size: 53

dsolve(diff(y(x),x$2)+a^2/y(x)=0,y(x), singsol=all)
 

\begin{align*} \int _{}^{y \left (x \right )}\frac {1}{\sqrt {-2 a^{2} \ln \left (\textit {\_a} \right )+c_{1}}}d \textit {\_a} -x -c_{2} &= 0 \\ -\int _{}^{y \left (x \right )}\frac {1}{\sqrt {-2 a^{2} \ln \left (\textit {\_a} \right )+c_{1}}}d \textit {\_a} -x -c_{2} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 60.116 (sec). Leaf size: 110

DSolve[D[y[x],{x,2}]+a^2/y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \exp \left (\frac {c_1}{2 a^2}-\text {erf}^{-1}\left (-\sqrt {\frac {2}{\pi }} \sqrt {a^2 e^{-\frac {c_1}{a^2}} (x+c_2){}^2}\right ){}^2\right ) \\ y(x)\to \exp \left (\frac {c_1}{2 a^2}-\text {erf}^{-1}\left (\sqrt {\frac {2}{\pi }} \sqrt {a^2 e^{-\frac {c_1}{a^2}} (x+c_2){}^2}\right ){}^2\right ) \\ \end{align*}