83.4.4 problem 4

Internal problem ID [19076]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter II. Equations of first order and first degree. Exercise II (C) at page 12
Problem number : 4
Date solved : Tuesday, January 28, 2025 at 12:52:00 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.198 (sec). Leaf size: 64

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

\[ y \left (x \right ) = \frac {x \left (\sqrt {3}\, \tan \left (\operatorname {RootOf}\left (\sqrt {3}\, \ln \left (\sec \left (\textit {\_Z} \right )^{6} \left (\sqrt {3}\, \sin \left (\textit {\_Z} \right )-3 \cos \left (\textit {\_Z} \right )\right )^{4} x^{6}\right )+\sqrt {3}\, \ln \left (3\right )-6 \sqrt {3}\, \ln \left (2\right )+6 c_{1} \sqrt {3}-6 \textit {\_Z} \right )\right )-1\right )}{2} \]

✓ Solution by Mathematica

Time used: 0.187 (sec). Leaf size: 86

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

\[ \text {Solve}\left [-\frac {\arctan \left (\frac {\frac {2 y(x)}{x}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (1-\frac {y(x)^3}{x^3}\right )-\frac {1}{6} \log \left (\frac {y(x)^2}{x^2}+\frac {y(x)}{x}+1\right )+\frac {1}{3} \log \left (1-\frac {y(x)}{x}\right )=-\log (x)+c_1,y(x)\right ] \]