83.28.7 problem 7

Internal problem ID [19372]
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 : Tuesday, January 28, 2025 at 01:35:41 PM
CAS classification : [[_2nd_order, _quadrature]]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 38

dsolve(diff(y(x),x$2)*sqrt(x^2+a^2)=x,y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {x \sqrt {a^{2}+x^{2}}}{2}+\frac {a^{2} \ln \left (x +\sqrt {a^{2}+x^{2}}\right )}{2}+c_{1} x +c_{2} \]

✓ Solution by Mathematica

Time used: 0.041 (sec). Leaf size: 49

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

\[ y(x)\to \frac {1}{2} a^2 \text {arctanh}\left (\frac {x}{\sqrt {a^2+x^2}}\right )+x \left (\frac {1}{2} \sqrt {a^2+x^2}+c_2\right )+c_1 \]