83.23.15 problem 15

Internal problem ID [19296]
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 : 15
Date solved : Tuesday, January 28, 2025 at 01:24:40 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

\begin{align*} {y^{\prime }}^{2}+2 x^{3} y^{\prime }-4 x^{2} y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.142 (sec). Leaf size: 19

dsolve(diff(y(x),x)^2+2*diff(y(x),x)*x^3-4*x^2*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.818 (sec). Leaf size: 142

DSolve[D[y[x],x]^2+2*D[y[x],x]*x^3-4*x^2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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