75.9.5 problem 224

Internal problem ID [16842]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 8.3. The Lagrange and Clairaut equations. Exercises page 72
Problem number : 224
Date solved : Tuesday, January 28, 2025 at 09:34:38 AM
CAS classification : [_dAlembert]

\begin{align*} y&=\frac {3 x y^{\prime }}{2}+{\mathrm e}^{y^{\prime }} \end{align*}

✓ Solution by Maple

Time used: 0.274 (sec). Leaf size: 201

dsolve(y(x)=3/2*x*diff(y(x),x)+exp(diff(y(x),x)),y(x), singsol=all)
 

\begin{align*} y &= 1 \\ \frac {27 x \left (\left (-2 x^{2} \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {2 y}{3 x}}}{3 x}\right )^{2}-4 x \left (x -\frac {2 y}{3}\right ) \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {2 y}{3 x}}}{3 x}\right )-4 x^{2}+\frac {8 x y}{3}-\frac {8 y^{2}}{9}\right ) {\mathrm e}^{\frac {-3 x \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {2 y}{3 x}}}{3 x}\right )+2 y}{3 x}}+\operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {2 y}{3 x}}}{3 x}\right )^{3} x^{3}-2 y \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {2 y}{3 x}}}{3 x}\right )^{2} x^{2}+\frac {4 y^{2} \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {2 y}{3 x}}}{3 x}\right ) x}{3}+\frac {c_{1} x^{2}}{27}-\frac {8 y^{3}}{27}\right )}{{\left (3 x \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {2 y}{3 x}}}{3 x}\right )-2 y\right )}^{3}} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.519 (sec). Leaf size: 52

DSolve[y[x]==3/2*x*D[y[x],x]+Exp[D[y[x],x]],y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\left \{x=-\frac {2 e^{K[1]} \left (K[1]^2-2 K[1]+2\right )}{K[1]^3}+\frac {c_1}{K[1]^3},y(x)=\frac {3}{2} x K[1]+e^{K[1]}\right \},\{y(x),K[1]\}\right ] \]