83.32.7 problem 7

Internal problem ID [19406]
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 (F) at page 113
Problem number : 7
Date solved : Tuesday, January 28, 2025 at 01:38:22 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} a^{2} {y^{\prime \prime }}^{2}&={y^{\prime }}^{2}+1 \end{align*}

✓ Solution by Maple

Time used: 0.614 (sec). Leaf size: 52

dsolve(a^2*diff(y(x),x$2)^2=1+diff(y(x),x)^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 {4 c_{2}^{2} {\mathrm e}^{\frac {x}{a}}+a^{2} {\mathrm e}^{-\frac {x}{a}}+4 c_{2} c_{1}}{4 c_{2}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.370 (sec). Leaf size: 40

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

\begin{align*} y(x)\to c_2-a \cosh \left (\frac {x}{a}-c_1\right ) \\ y(x)\to a \cosh \left (\frac {x}{a}+c_1\right )+c_2 \\ \end{align*}