60.1.338 problem 339

Internal problem ID [10352]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 339
Date solved : Monday, January 27, 2025 at 07:21:59 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

\begin{align*} \left (x \sqrt {1+x^{2}+y^{2}}-y \left (x^{2}+y^{2}\right )\right ) y^{\prime }-y \sqrt {1+x^{2}+y^{2}}-x \left (x^{2}+y^{2}\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.115 (sec). Leaf size: 25

dsolve((x*(x^2+y(x)^2+1)^(1/2)-y(x)*(y(x)^2+x^2))*diff(y(x),x)-y(x)*(x^2+y(x)^2+1)^(1/2)-x*(y(x)^2+x^2) = 0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.520 (sec). Leaf size: 661

DSolve[-(x*(x^2 + y[x]^2)) - y[x]*Sqrt[1 + x^2 + y[x]^2] + (-(y[x]*(x^2 + y[x]^2)) + x*Sqrt[1 + x^2 + y[x]^2])*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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