64.3.7 problem 8

Internal problem ID [13278]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 2, section 2.1 (Exact differential equations and integrating factors). Exercises page 37
Problem number : 8
Date solved : Tuesday, January 28, 2025 at 05:14:12 AM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

\begin{align*} \frac {x}{y^{2}}+x +\left (\frac {x^{2}}{y^{3}}+y\right ) y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.115 (sec). Leaf size: 52

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

\[ \frac {\left (-2 y^{2}-2\right ) \ln \left (1+y^{2}\right )+y^{4}+\left (x^{2}+2 c_{1} +1\right ) y^{2}+2 c_{1} -1}{2 y^{2}+2} = 0 \]

✓ Solution by Mathematica

Time used: 0.272 (sec). Leaf size: 124

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

\[ \text {Solve}\left [\frac {1}{2} x^2 y(x)^3 \exp \left (\int _1^{y(x)}\frac {1-3 K[1]^2}{K[1]^3+K[1]}dK[1]\right )+\frac {1}{2} x^2 y(x) \exp \left (\int _1^{y(x)}\frac {1-3 K[1]^2}{K[1]^3+K[1]}dK[1]\right )+\int _1^{y(x)}\exp \left (\int _1^{K[2]}\frac {1-3 K[1]^2}{K[1]^3+K[1]}dK[1]\right ) K[2]^4dK[2]=c_1,y(x)\right ] \]