83.23.28 problem 28

Internal problem ID [19309]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter V. Singular solutions. Exercise V at page 76
Problem number : 28
Date solved : Tuesday, January 28, 2025 at 01:30:46 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _dAlembert]

\begin{align*} 8 {y^{\prime }}^{3} x&=y \left (12 {y^{\prime }}^{2}-9\right ) \end{align*}

✓ Solution by Maple

Time used: 0.048 (sec). Leaf size: 61

dsolve(8*diff(y(x),x)^3*x=y(x)*(12*diff(y(x),x)^2-9),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.331 (sec). Leaf size: 77

DSolve[8*D[y[x],x]^3*x==y[x]*(12*D[y[x],x]^2-9),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {(x+3 c_1){}^{3/2}}{3 \sqrt {c_1}} \\ y(x)\to \frac {(x+3 c_1){}^{3/2}}{3 \sqrt {c_1}} \\ y(x)\to 0 \\ y(x)\to \text {Indeterminate} \\ y(x)\to -\frac {3 x}{2} \\ y(x)\to \frac {3 x}{2} \\ \end{align*}