60.2.118 problem 694

Internal problem ID [10705]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 694
Date solved : Tuesday, January 28, 2025 at 05:06:19 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} y^{\prime }&=\frac {x +1+2 \sqrt {4 x^{2} y+1}\, x^{3}}{2 x^{3} \left (1+x \right )} \end{align*}

✓ Solution by Maple

Time used: 0.345 (sec). Leaf size: 36

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

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

✓ Solution by Mathematica

Time used: 0.387 (sec). Leaf size: 640

DSolve[D[y[x],x] == (1/2 + x/2 + x^3*Sqrt[1 + 4*x^2*y[x]])/(x^3*(1 + x)),y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\left (-\frac {\exp \left (-\int _1^x\left (\frac {1}{K[1]+1}-\frac {1}{K[1]}\right )dK[1]\right ) x}{\sqrt {4 K[3] x^2+1}}-\int _1^x\left (-\frac {16 \exp \left (-\int _1^{K[2]}\left (\frac {1}{K[1]+1}-\frac {1}{K[1]}\right )dK[1]\right ) K[3]^2 K[2]^3}{\left (4 K[3] K[2]^2+1\right )^{3/2}}+\frac {4 \exp \left (-\int _1^{K[2]}\left (\frac {1}{K[1]+1}-\frac {1}{K[1]}\right )dK[1]\right ) K[3] K[2]^2}{\left (4 K[3] K[2]^2+1\right )^{3/2}}+\frac {12 \exp \left (-\int _1^{K[2]}\left (\frac {1}{K[1]+1}-\frac {1}{K[1]}\right )dK[1]\right ) K[3] K[2]}{\sqrt {4 K[3] K[2]^2+1}}-\frac {\exp \left (-\int _1^{K[2]}\left (\frac {1}{K[1]+1}-\frac {1}{K[1]}\right )dK[1]\right )}{\sqrt {4 K[3] K[2]^2+1}}-\frac {2 \exp \left (-\int _1^{K[2]}\left (\frac {1}{K[1]+1}-\frac {1}{K[1]}\right )dK[1]\right ) \sqrt {4 K[3] K[2]^2+1}}{K[2]}+\frac {\exp \left (-\int _1^{K[2]}\left (\frac {1}{K[1]+1}-\frac {1}{K[1]}\right )dK[1]\right )}{\sqrt {4 K[3] K[2]^2+1} K[2]}\right )dK[2]-\frac {\exp \left (-\int _1^x\left (\frac {1}{K[1]+1}-\frac {1}{K[1]}\right )dK[1]\right )}{\sqrt {4 K[3] x^2+1}}\right )dK[3]+\int _1^x\left (\frac {8 \exp \left (-\int _1^{K[2]}\left (\frac {1}{K[1]+1}-\frac {1}{K[1]}\right )dK[1]\right ) K[2] y(x)^2}{\sqrt {4 y(x) K[2]^2+1}}-\frac {2 \exp \left (-\int _1^{K[2]}\left (\frac {1}{K[1]+1}-\frac {1}{K[1]}\right )dK[1]\right ) \sqrt {4 y(x) K[2]^2+1} y(x)}{K[2]}-\frac {2 \exp \left (-\int _1^{K[2]}\left (\frac {1}{K[1]+1}-\frac {1}{K[1]}\right )dK[1]\right ) y(x)}{\sqrt {4 y(x) K[2]^2+1}}+\exp \left (-\int _1^{K[2]}\left (\frac {1}{K[1]+1}-\frac {1}{K[1]}\right )dK[1]\right )+\frac {\exp \left (-\int _1^{K[2]}\left (\frac {1}{K[1]+1}-\frac {1}{K[1]}\right )dK[1]\right ) \sqrt {4 y(x) K[2]^2+1}}{2 K[2]^2}+\frac {\exp \left (-\int _1^{K[2]}\left (\frac {1}{K[1]+1}-\frac {1}{K[1]}\right )dK[1]\right ) \sqrt {4 y(x) K[2]^2+1}}{2 K[2]^3}\right )dK[2]=c_1,y(x)\right ] \]