60.2.302 problem 879

Internal problem ID [10889]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 879
Date solved : Tuesday, January 28, 2025 at 05:29:12 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

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

✓ Solution by Maple

Time used: 0.114 (sec). Leaf size: 56

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

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

✓ Solution by Mathematica

Time used: 37.232 (sec). Leaf size: 220

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