60.2.286 problem 863

Internal problem ID [10873]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 863
Date solved : Tuesday, January 28, 2025 at 05:27:34 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} y^{\prime }&=\frac {y+x \sqrt {x^{2}+y^{2}}+x^{3} \sqrt {x^{2}+y^{2}}+x^{4} \sqrt {x^{2}+y^{2}}}{x} \end{align*}

✓ Solution by Maple

Time used: 1.595 (sec). Leaf size: 38

dsolve(diff(y(x),x) = (y(x)+x*(y(x)^2+x^2)^(1/2)+x^3*(y(x)^2+x^2)^(1/2)+x^4*(y(x)^2+x^2)^(1/2))/x,y(x), singsol=all)
 

\[ \ln \left (\sqrt {x^{2}+y^{2}}+y\right )-\frac {x^{4}}{4}-\frac {x^{3}}{3}-x -\ln \left (x \right )-c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 0.378 (sec). Leaf size: 26

DSolve[D[y[x],x] == (y[x] + x*Sqrt[x^2 + y[x]^2] + x^3*Sqrt[x^2 + y[x]^2] + x^4*Sqrt[x^2 + y[x]^2])/x,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to x \sinh \left (\frac {x^4}{4}+\frac {x^3}{3}+x+c_1\right ) \]