83.35.13 problem 13

Internal problem ID [19433]
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 : 13
Date solved : Tuesday, January 28, 2025 at 01:42:17 PM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Solution by Maple

Time used: 0.362 (sec). Leaf size: 89

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

\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= c_{1} \\ y \left (x \right ) &= \frac {a \left ({\mathrm e}^{\frac {\sqrt {c_{1}^{2}-1}\, \left (c_{2} +x \right )}{a}}-c_{1} \right )}{\sqrt {c_{1}^{2}-1}} \\ y \left (x \right ) &= c_{1} x +c_{2} \\ y \left (x \right ) &= c_{1} {\mathrm e}^{\frac {x}{a}}+c_{2} {\mathrm e}^{-\frac {x}{a}} \\ y \left (x \right ) &= c_{1} \sin \left (\frac {x}{a}\right )+c_{2} \cos \left (\frac {x}{a}\right ) \\ \end{align*}

✗ Solution by Mathematica

Time used: 0.000 (sec). Leaf size: 0

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

Not solved