56.1.23 problem 24

Internal problem ID [8735]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 24
Date solved : Monday, January 27, 2025 at 04:45:56 PM
CAS classification : [_separable]

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

✓ Solution by Maple

Time used: 0.031 (sec). Leaf size: 97

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

\begin{align*} \ln \left (x \right )-\sqrt {4 y+1}-\frac {\ln \left (-1+\sqrt {4 y+1}\right )}{2}+\frac {\ln \left (1+\sqrt {4 y+1}\right )}{2}-\frac {\ln \left (y\right )}{2}-c_{1} &= 0 \\ \ln \left (x \right )+\sqrt {4 y+1}+\frac {\ln \left (-1+\sqrt {4 y+1}\right )}{2}-\frac {\ln \left (1+\sqrt {4 y+1}\right )}{2}-\frac {\ln \left (y\right )}{2}-c_{1} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 19.224 (sec). Leaf size: 72

DSolve[y[x]==x*D[y[x],x]+x^2*(D[y[x],x])^2,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {1}{4} W\left (-e^{-1-2 c_1} x\right ) \left (2+W\left (-e^{-1-2 c_1} x\right )\right ) \\ y(x)\to \frac {1}{4} W\left (e^{-1+2 c_1} x\right ) \left (2+W\left (e^{-1+2 c_1} x\right )\right ) \\ y(x)\to 0 \\ \end{align*}