60.2.298 problem 875

Internal problem ID [10885]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 875
Date solved : Tuesday, January 28, 2025 at 05:28:32 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} y^{\prime }&=-\frac {-y x -y+x^{5} \sqrt {x^{2}+y^{2}}-x^{4} \sqrt {x^{2}+y^{2}}\, y}{x \left (1+x \right )} \end{align*}

Solution by Maple

Time used: 0.234 (sec). Leaf size: 74

dsolve(diff(y(x),x) = -(-x*y(x)-y(x)+x^5*(y(x)^2+x^2)^(1/2)-x^4*(y(x)^2+x^2)^(1/2)*y(x))/x/(x+1),y(x), singsol=all)
 
\[ \ln \left (\frac {x \left (\sqrt {2 x^{2}+2 y^{2}}+y+x \right )}{y-x}\right )+\sqrt {2}\, \ln \left (x +1\right )+\frac {\left (3 x^{4}-4 x^{3}+6 x^{2}-12 x \right ) \sqrt {2}}{12}-c_{1} +\ln \left (2\right )-\ln \left (x \right ) = 0 \]

Solution by Mathematica

Time used: 37.175 (sec). Leaf size: 232

DSolve[D[y[x],x] == (y[x] + x*y[x] - x^5*Sqrt[x^2 + y[x]^2] + x^4*y[x]*Sqrt[x^2 + y[x]^2])/(x*(1 + x)),y[x],x,IncludeSingularSolutions -> True]
 
\begin{align*} y(x)\to \frac {x-2 \sqrt {x^2 \tanh ^2\left (\sqrt {2} \left (\int _1^x\frac {K[1]^4}{K[1]+1}dK[1]+c_1\right )\right ) \text {sech}^2\left (\sqrt {2} \left (\int _1^x\frac {K[1]^4}{K[1]+1}dK[1]+c_1\right )\right )}}{-1+2 \tanh ^2\left (\sqrt {2} \left (\int _1^x\frac {K[1]^4}{K[1]+1}dK[1]+c_1\right )\right )} \\ y(x)\to \frac {x+2 \sqrt {x^2 \tanh ^2\left (\sqrt {2} \left (\int _1^x\frac {K[1]^4}{K[1]+1}dK[1]+c_1\right )\right ) \text {sech}^2\left (\sqrt {2} \left (\int _1^x\frac {K[1]^4}{K[1]+1}dK[1]+c_1\right )\right )}}{-1+2 \tanh ^2\left (\sqrt {2} \left (\int _1^x\frac {K[1]^4}{K[1]+1}dK[1]+c_1\right )\right )} \\ y(x)\to x \\ \end{align*}