83.32.9 problem 9

Internal problem ID [19408]
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 : 9
Date solved : Tuesday, January 28, 2025 at 08:30:20 PM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]

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

✓ Solution by Maple

Time used: 0.063 (sec). Leaf size: 34

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

\[ y \left (x \right ) = \int \operatorname {RootOf}\left (-\textit {\_Z}^{2}-\textit {\_Z} \sqrt {\textit {\_Z}^{2}+1}-\operatorname {arcsinh}\left (\textit {\_Z} \right )-2 \ln \left (x \right )+c_{1} \right )d x +c_{2} \]

✓ Solution by Mathematica

Time used: 0.176 (sec). Leaf size: 64

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

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