60.2.95 problem 671

Internal problem ID [10682]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 671
Date solved : Tuesday, January 28, 2025 at 05:04:56 PM
CAS classification : [_rational]

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

✓ Solution by Maple

Time used: 0.045 (sec). Leaf size: 197

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

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

✓ Solution by Mathematica

Time used: 0.222 (sec). Leaf size: 144

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

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