62.38.1 problem Ex 1

Internal problem ID [13012]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IX, Miscellaneous methods for solving equations of higher order than first. Article 62. Summary. Page 144
Problem number : Ex 1
Date solved : Tuesday, January 28, 2025 at 04:49:09 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]

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

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 17

dsolve(diff(y(x),x$2)=diff(y(x),x)^2+1,y(x), singsol=all)
 

\[ y = -\ln \left (\sin \left (x \right ) c_{1} -\cos \left (x \right ) c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.800 (sec). Leaf size: 38

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

\[ y(x)\to \int _1^x\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1]^2+1}dK[1]\&\right ][c_1+K[2]]dK[2]+c_2 \]