75.8.16 problem 214

Internal problem ID [16832]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 8. First order not solved for the derivative. Exercises page 67
Problem number : 214
Date solved : Tuesday, January 28, 2025 at 09:34:18 AM
CAS classification : [_quadrature]

\begin{align*} {y^{\prime }}^{2} x&={\mathrm e}^{\frac {1}{y^{\prime }}} \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 71

dsolve(diff(y(x),x)^2*x=exp(1/diff(y(x),x)),y(x), singsol=all)
 

\begin{align*} y &= \frac {4 c_{1} \operatorname {LambertW}\left (-\frac {\sqrt {x}}{2}\right )^{2}+2 x \operatorname {LambertW}\left (-\frac {\sqrt {x}}{2}\right )+x}{4 \operatorname {LambertW}\left (-\frac {\sqrt {x}}{2}\right )^{2}} \\ y &= \frac {4 c_{1} \operatorname {LambertW}\left (\frac {\sqrt {x}}{2}\right )^{2}+2 x \operatorname {LambertW}\left (\frac {\sqrt {x}}{2}\right )+x}{4 \operatorname {LambertW}\left (\frac {\sqrt {x}}{2}\right )^{2}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.034 (sec). Leaf size: 67

DSolve[D[y[x],x]^2*x==Exp[1/D[y[x],x]],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \int _1^x\frac {1}{2 W\left (-\frac {1}{2 \sqrt {\frac {1}{K[1]}}}\right )}dK[1]+c_1 \\ y(x)\to \int _1^x\frac {1}{2 W\left (\frac {1}{2 \sqrt {\frac {1}{K[2]}}}\right )}dK[2]+c_1 \\ \end{align*}