83.32.6 problem 6

Internal problem ID [19405]
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 : 6
Date solved : Tuesday, January 28, 2025 at 01:38:06 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]

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

✓ Solution by Maple

Time used: 0.012 (sec). Leaf size: 31

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

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

✓ Solution by Mathematica

Time used: 0.242 (sec). Leaf size: 22

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

\[ y(x)\to c_2-\frac {\log (\cos (a k (x+c_1)))}{k^2} \]