60.2.249 problem 825

Internal problem ID [10836]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 825
Date solved : Tuesday, January 28, 2025 at 05:22:08 PM
CAS classification : [_Abel]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 48

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

\[ y = \frac {\left (19 \operatorname {RootOf}\left (-1296 \left (\int _{}^{\textit {\_Z}}\frac {1}{361 \textit {\_a}^{3}-432 \textit {\_a} +432}d \textit {\_a} \right )+2 \ln \left (x^{2}+1\right )+3 c_{1} \right )-6\right ) \sqrt {x^{2}+1}}{18} \]

✓ Solution by Mathematica

Time used: 1.518 (sec). Leaf size: 129

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

\[ \text {Solve}\left [\int _1^{\frac {\frac {3 y(x) x}{\left (x^2+1\right )^2}+\frac {x}{\left (x^2+1\right )^{3/2}}}{\sqrt [3]{38} \sqrt [3]{\frac {x^3}{\left (x^2+1\right )^{9/2}}}}}\frac {1}{K[1]^3-\frac {6 \sqrt [3]{2} K[1]}{19^{2/3}}+1}dK[1]=\frac {19^{2/3} \left (\frac {x^3}{\left (x^2+1\right )^{9/2}}\right )^{2/3} \left (x^2+1\right )^3 \log \left (x^2+1\right )}{9 \sqrt [3]{2} x^2}+c_1,y(x)\right ] \]