62.20.2 problem Ex 2

Internal problem ID [12916]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter V, Singular solutions. Article 33. Page 73
Problem number : Ex 2
Date solved : Tuesday, January 28, 2025 at 04:43:24 AM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

\begin{align*} 8 \left (1+y^{\prime }\right )^{3}&=27 \left (x +y\right ) \left (1-y^{\prime }\right )^{3} \end{align*}

✓ Solution by Maple

Time used: 0.690 (sec). Leaf size: 138

dsolve(8*(1+diff(y(x),x))^3=27*(x+y(x))*(1-diff(y(x),x))^3,y(x), singsol=all)
 

\begin{align*} y &= -x \\ \frac {x}{2}-\frac {4 \ln \left (27 y+27 x +8\right )}{27}+\frac {4 \ln \left (9 \left (x +y\right )^{{2}/{3}}-6 \left (x +y\right )^{{1}/{3}}+4\right )}{27}+\frac {4 \ln \left (2+3 \left (x +y\right )^{{1}/{3}}\right )}{27}-\frac {y}{2}-\frac {\left (x +y\right )^{{2}/{3}}}{2}-c_{1} &= 0 \\ \frac {x}{2}-\frac {y}{2}-\frac {i \sqrt {3}\, \left (x +y\right )^{{2}/{3}}}{4}+\frac {\left (x +y\right )^{{2}/{3}}}{4}-c_{1} &= 0 \\ \frac {x}{2}-\frac {y}{2}+\frac {\left (x +y\right )^{{2}/{3}}}{4}+\frac {i \sqrt {3}\, \left (x +y\right )^{{2}/{3}}}{4}-c_{1} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 178.922 (sec). Leaf size: 58354

DSolve[8*(1+D[y[x],x])^3==27*(x+y[x])*(1-D[y[x],x])^3,y[x],x,IncludeSingularSolutions -> True]
 

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