54.2.22 problem 25

Internal problem ID [8554]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 16. Nonlinear equations. Miscellaneous Exercises. Page 340
Problem number : 25
Date solved : Monday, January 27, 2025 at 04:14:09 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} y {y^{\prime }}^{2}-\left (x +y\right ) y^{\prime }+y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.061 (sec). Leaf size: 176

dsolve(y(x)*diff(y(x),x)^2-(x+y(x))*diff(y(x),x)+y(x)=0,y(x), singsol=all)
 

\begin{align*} y &= x \\ y &= 0 \\ \frac {-x \sqrt {\frac {\left (3 y+x \right ) \left (x -y\right )}{x^{2}}}+2 y \ln \left (\frac {y}{x}\right )+\left (-2 \,\operatorname {arctanh}\left (\frac {x +y}{x \sqrt {\frac {\left (3 y+x \right ) \left (x -y\right )}{x^{2}}}}\right )-2 c_{1} +2 \ln \left (x \right )\right ) y-x}{2 y} &= 0 \\ \frac {x \sqrt {\frac {\left (3 y+x \right ) \left (x -y\right )}{x^{2}}}+2 y \ln \left (\frac {y}{x}\right )+\left (2 \,\operatorname {arctanh}\left (\frac {x +y}{x \sqrt {\frac {\left (3 y+x \right ) \left (x -y\right )}{x^{2}}}}\right )-2 c_{1} +2 \ln \left (x \right )\right ) y-x}{2 y} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.901 (sec). Leaf size: 192

DSolve[y[x]*D[y[x],x]^2-(x+y[x])*D[y[x],x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} \text {Solve}\left [\log \left (\sqrt {\frac {y(x)}{x}-1}+i \sqrt {\frac {3 y(x)}{x}+1}\right )-\frac {i \sqrt {\frac {3 y(x)}{x}+1}}{\sqrt {\frac {y(x)}{x}-1}+i \sqrt {\frac {3 y(x)}{x}+1}}&=-\frac {\log (x)}{2}+c_1,y(x)\right ] \\ \text {Solve}\left [\log \left (\sqrt {\frac {y(x)}{x}-1}-i \sqrt {\frac {3 y(x)}{x}+1}\right )-\frac {\sqrt {\frac {3 y(x)}{x}+1}}{\sqrt {\frac {3 y(x)}{x}+1}+i \sqrt {\frac {y(x)}{x}-1}}&=-\frac {\log (x)}{2}+c_1,y(x)\right ] \\ y(x)\to 0 \\ \end{align*}