83.35.8 problem 8

Internal problem ID [19428]
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. Misc. Exercise on chapter VII. Page 118
Problem number : 8
Date solved : Tuesday, January 28, 2025 at 01:39:36 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} -a y^{\prime \prime }&=\left ({y^{\prime }}^{2}+1\right )^{{3}/{2}} \end{align*}

✓ Solution by Maple

Time used: 0.069 (sec). Leaf size: 91

dsolve(-a*diff(y(x),x$2)=(1+diff(y(x),x)^2)^(3/2),y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= -i x +c_{1} \\ y \left (x \right ) &= i x +c_{1} \\ y \left (x \right ) &= \frac {\left (a +x +c_{1} \right ) \left (-a +x +c_{1} \right )}{\sqrt {a^{2}-c_{1}^{2}-2 c_{1} x -x^{2}}}+c_{2} \\ y \left (x \right ) &= \frac {\left (a +x +c_{1} \right ) \left (a -x -c_{1} \right )}{\sqrt {a^{2}-c_{1}^{2}-2 c_{1} x -x^{2}}}+c_{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.591 (sec). Leaf size: 71

DSolve[-a*D[y[x],{x,2}]==(1+D[y[x],x]^2)^(3/2),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to c_2-i \sqrt {a^2 \left (-1+c_1{}^2\right )-2 a c_1 x+x^2} \\ y(x)\to i \sqrt {a^2 \left (-1+c_1{}^2\right )-2 a c_1 x+x^2}+c_2 \\ \end{align*}