15.19.22 problem 22

Internal problem ID [3306]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 37, page 171
Problem number : 22
Date solved : Monday, January 27, 2025 at 07:32:53 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} \left (1+{y^{\prime }}^{2}\right ) x&=\left (x +y\right ) y^{\prime } \end{align*}

✓ Solution by Maple

Time used: 0.029 (sec). Leaf size: 36

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

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

✓ Solution by Mathematica

Time used: 5.364 (sec). Leaf size: 247

DSolve[(D[y[x],x]^2+1)*x==D[y[x],x]*(x+y[x]),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} \text {Solve}\left [2 \left (\text {arctanh}\left (\frac {\sqrt {\frac {y(x)}{x}+3}-2}{\sqrt {\frac {y(x)}{x}-1}}\right )+\frac {\sqrt {\frac {y(x)}{x}-1} \left (\sqrt {\frac {y(x)}{x}+3}-2\right )}{\frac {2 y(x)}{x}-2 \sqrt {\frac {y(x)}{x}-1} \left (\sqrt {\frac {y(x)}{x}+3}-2\right )-4 \sqrt {\frac {y(x)}{x}+3}+6}\right )&=\frac {\log (x)}{2}+c_1,y(x)\right ] \\ \text {Solve}\left [2 \text {arctanh}\left (\frac {\sqrt {\frac {y(x)}{x}+3}-2}{\sqrt {\frac {y(x)}{x}-1}}\right )+\frac {\sqrt {\frac {y(x)}{x}-1} \left (\sqrt {\frac {y(x)}{x}+3}-2\right )}{\frac {y(x)}{x}+\sqrt {\frac {y(x)}{x}-1} \left (\sqrt {\frac {y(x)}{x}+3}-2\right )-2 \sqrt {\frac {y(x)}{x}+3}+3}&=-\frac {\log (x)}{2}+c_1,y(x)\right ] \\ \end{align*}