60.2.164 problem 740

Internal problem ID [10751]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 740
Date solved : Tuesday, January 28, 2025 at 05:10:34 PM
CAS classification : [_rational]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 72

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

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

✓ Solution by Mathematica

Time used: 0.128 (sec). Leaf size: 159

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

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