29.36.15 problem 1081

Internal problem ID [5642]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 36
Problem number : 1081
Date solved : Monday, January 27, 2025 at 12:51:06 PM
CAS classification : [`y=_G(x,y')`]

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

✗ Solution by Maple

dsolve(x*y(x)^2*diff(y(x),x)^3-y(x)^3*diff(y(x),x)^2+x*(x^2+1)*diff(y(x),x)-x^2*y(x) = 0,y(x), singsol=all)
 

\[ \text {No solution found} \]

✓ Solution by Mathematica

Time used: 0.544 (sec). Leaf size: 399

DSolve[x y[x]^2 (D[y[x],x])^3 -y[x]^3 (D[y[x],x])^2 + x (1+x^2) D[y[x],x] -x^2 y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\sqrt {c_1 \left (x^2+\frac {1}{1+c_1{}^2}\right )} \\ y(x)\to \sqrt {c_1 \left (x^2+\frac {1}{1+c_1{}^2}\right )} \\ y(x)\to -\frac {\sqrt [4]{-8 x^4+20 x^2-\sqrt {-\left (8 x^2-1\right )^3}+1}}{2^{3/4}} \\ y(x)\to -\frac {i \sqrt [4]{-8 x^4+20 x^2-\sqrt {-\left (8 x^2-1\right )^3}+1}}{2^{3/4}} \\ y(x)\to \frac {i \sqrt [4]{-8 x^4+20 x^2-\sqrt {-\left (8 x^2-1\right )^3}+1}}{2^{3/4}} \\ y(x)\to \frac {\sqrt [4]{-8 x^4+20 x^2-\sqrt {-\left (8 x^2-1\right )^3}+1}}{2^{3/4}} \\ y(x)\to -\frac {\sqrt [4]{-8 x^4+20 x^2+\sqrt {-\left (8 x^2-1\right )^3}+1}}{2^{3/4}} \\ y(x)\to -\frac {i \sqrt [4]{-8 x^4+20 x^2+\sqrt {-\left (8 x^2-1\right )^3}+1}}{2^{3/4}} \\ y(x)\to \frac {i \sqrt [4]{-8 x^4+20 x^2+\sqrt {-\left (8 x^2-1\right )^3}+1}}{2^{3/4}} \\ y(x)\to \frac {\sqrt [4]{-8 x^4+20 x^2+\sqrt {-\left (8 x^2-1\right )^3}+1}}{2^{3/4}} \\ \end{align*}