77.28.7 problem 7

Internal problem ID [20657]
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 (B) at page 106
Problem number : 7
Date solved : Thursday, October 02, 2025 at 06:18:00 PM
CAS classification : [[_2nd_order, _quadrature]]

\begin{align*} y^{\prime \prime } \sqrt {a^{2}+x^{2}}&=x \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 38

ode:=diff(diff(y(x),x),x)*(a^2+x^2)^(1/2) = x; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {x \sqrt {a^{2}+x^{2}}}{2}+\frac {a^{2} \ln \left (x +\sqrt {a^{2}+x^{2}}\right )}{2}+c_1 x +c_2 \]

✓ Mathematica. Time used: 0.035 (sec). Leaf size: 49

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

\begin{align*} y(x)&\to \frac {1}{2} a^2 \text {arctanh}\left (\frac {x}{\sqrt {a^2+x^2}}\right )+x \left (\frac {1}{2} \sqrt {a^2+x^2}+c_2\right )+c_1 \end{align*}

✓ Sympy. Time used: 0.640 (sec). Leaf size: 46

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

\[ y{\left (x \right )} = C_{1} + C_{2} x + \frac {a^{2} \operatorname {asinh}{\left (\frac {x}{a} \right )}}{2} - \frac {a x \sqrt {1 + \frac {x^{2}}{a^{2}}}}{2} + x \sqrt {a^{2} + x^{2}} \]