82.42.4 problem Ex. 4

Internal problem ID [18889]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter VIII. Exact differential equations, and equations of particular forms. Integration in series. problems at page 97
Problem number : Ex. 4
Date solved : Monday, March 31, 2025 at 06:19:44 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

✓ Maple. Time used: 0.085 (sec). Leaf size: 269

ode:=diff(diff(y(x),x),x)-a^2/y(x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {c_1 \left ({\mathrm e}^{\operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{4} a^{4}-2 \textit {\_Z} \,c_1^{3} a^{2} {\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{2}-2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) c_2 -2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) x \right )}+2 a^{2} c_1 +{\mathrm e}^{-\operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{4} a^{4}-2 \textit {\_Z} \,c_1^{3} a^{2} {\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{2}-2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) c_2 -2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) x \right )} c_1^{2} a^{4}\right )}{2} \\ y &= \frac {c_1 \left (2 a^{2} c_1 +{\mathrm e}^{\operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{4} a^{4}-2 \textit {\_Z} \,c_1^{3} a^{2} {\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{2}+2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) c_2 +2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) x \right )}+{\mathrm e}^{-\operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{4} a^{4}-2 \textit {\_Z} \,c_1^{3} a^{2} {\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{2 \textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) c_1^{2}+2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) c_2 +2 \,{\mathrm e}^{\textit {\_Z}} \operatorname {csgn}\left (\frac {1}{c_1}\right ) x \right )} c_1^{2} a^{4}\right )}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.173 (sec). Leaf size: 71

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

\[ \text {Solve}\left [\left (\frac {2 a^2 \text {arctanh}\left (\frac {\sqrt {-\frac {2 a^2}{y(x)}+c_1}}{\sqrt {c_1}}\right )}{c_1{}^{3/2}}+\frac {y(x) \sqrt {-\frac {2 a^2}{y(x)}+c_1}}{c_1}\right ){}^2=(x+c_2){}^2,y(x)\right ] \]

✓ Sympy. Time used: 2.237 (sec). Leaf size: 405

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a**2/y(x)**2 + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ \begin {cases} - \frac {\sqrt {2} C_{1} a \sqrt {y{\left (x \right )}}}{\sqrt {\frac {C_{1} y{\left (x \right )}}{2 a^{2}} - 1}} + \frac {\sqrt {2} y^{\frac {3}{2}}{\left (x \right )}}{2 a \sqrt {\frac {C_{1} y{\left (x \right )}}{2 a^{2}} - 1}} + \frac {2 a^{2} \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {C_{1}} \sqrt {y{\left (x \right )}}}{2 a} \right )}}{C_{1}^{\frac {3}{2}}} & \text {for}\: \left |{\frac {C_{1} y{\left (x \right )}}{a^{2}}}\right | > 2 \\\frac {\sqrt {2} i C_{1} a \sqrt {y{\left (x \right )}}}{\sqrt {- \frac {C_{1} y{\left (x \right )}}{2 a^{2}} + 1}} - \frac {\sqrt {2} i y^{\frac {3}{2}}{\left (x \right )}}{2 a \sqrt {- \frac {C_{1} y{\left (x \right )}}{2 a^{2}} + 1}} - \frac {2 i a^{2} \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {C_{1}} \sqrt {y{\left (x \right )}}}{2 a} \right )}}{C_{1}^{\frac {3}{2}}} & \text {otherwise} \end {cases} = C_{1} + x, \ \begin {cases} - \frac {\sqrt {2} C_{1} a \sqrt {y{\left (x \right )}}}{\sqrt {\frac {C_{1} y{\left (x \right )}}{2 a^{2}} - 1}} + \frac {\sqrt {2} y^{\frac {3}{2}}{\left (x \right )}}{2 a \sqrt {\frac {C_{1} y{\left (x \right )}}{2 a^{2}} - 1}} + \frac {2 a^{2} \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {C_{1}} \sqrt {y{\left (x \right )}}}{2 a} \right )}}{C_{1}^{\frac {3}{2}}} & \text {for}\: \left |{\frac {C_{1} y{\left (x \right )}}{a^{2}}}\right | > 2 \\\frac {\sqrt {2} i C_{1} a \sqrt {y{\left (x \right )}}}{\sqrt {- \frac {C_{1} y{\left (x \right )}}{2 a^{2}} + 1}} - \frac {\sqrt {2} i y^{\frac {3}{2}}{\left (x \right )}}{2 a \sqrt {- \frac {C_{1} y{\left (x \right )}}{2 a^{2}} + 1}} - \frac {2 i a^{2} \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {C_{1}} \sqrt {y{\left (x \right )}}}{2 a} \right )}}{C_{1}^{\frac {3}{2}}} & \text {otherwise} \end {cases} = C_{1} - x\right ] \]