60.1.540 problem 543

Internal problem ID [10554]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 543
Date solved : Monday, January 27, 2025 at 08:58:21 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: 0.546 (sec). Leaf size: 399

DSolve[-(x^2*y[x]) + x*(1 + x^2)*D[y[x],x] - y[x]^3*D[y[x],x]^2 + x*y[x]^2*D[y[x],x]^3==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*}