60.1.237 problem 238

Internal problem ID [10251]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 238
Date solved : Monday, January 27, 2025 at 06:41:43 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational, [_Abel, `2nd type`, `class B`]]

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

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 91

dsolve((x*(y(x)+x)+a)*diff(y(x),x)-y(x)*(y(x)+x)-b=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 2.794 (sec). Leaf size: 1598

DSolve[(x*(y[x]+x)+a)*D[y[x],x]-y[x]*(y[x]+x)-b==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {-\frac {1}{\frac {a}{a^2+a x^2+b x^2}-\frac {\exp \left (\int _1^x\frac {a^2-2 K[1]^2 a-2 b K[1]^2}{a K[1]^3+b K[1]^3+a^2 K[1]}dK[1]\right )}{\sqrt {-2 \int _1^x-\frac {\exp \left (2 \int _1^{K[2]}\frac {a^2-2 K[1]^2 a-2 b K[1]^2}{a K[1]^3+b K[1]^3+a^2 K[1]}dK[1]\right ) \left (a^2+K[2]^2 a+b K[2]^2\right )}{K[2]}dK[2]+c_1}}}+a+x^2}{x} \\ y(x)\to -\frac {-\frac {1}{\frac {a}{a^2+a x^2+b x^2}+\frac {\exp \left (\int _1^x\frac {a^2-2 K[1]^2 a-2 b K[1]^2}{a K[1]^3+b K[1]^3+a^2 K[1]}dK[1]\right )}{\sqrt {-2 \int _1^x-\frac {\exp \left (2 \int _1^{K[2]}\frac {a^2-2 K[1]^2 a-2 b K[1]^2}{a K[1]^3+b K[1]^3+a^2 K[1]}dK[1]\right ) \left (a^2+K[2]^2 a+b K[2]^2\right )}{K[2]}dK[2]+c_1}}}+a+x^2}{x} \\ y(x)\to \frac {b x}{a} \\ y(x)\to -\frac {2 \sqrt {2} a^2 x^2 \exp \left (\int _1^x\frac {a^2-2 K[1]^2 a-2 b K[1]^2}{a K[1]^3+b K[1]^3+a^2 K[1]}dK[1]\right )+\sqrt {2} a x^2 \left (b+x^2\right ) \exp \left (\int _1^x\frac {a^2-2 K[1]^2 a-2 b K[1]^2}{a K[1]^3+b K[1]^3+a^2 K[1]}dK[1]\right )+b \left (\sqrt {2} x^4 \exp \left (\int _1^x\frac {a^2-2 K[1]^2 a-2 b K[1]^2}{a K[1]^3+b K[1]^3+a^2 K[1]}dK[1]\right )-2 x^2 \sqrt {-\int _1^x-\frac {\exp \left (2 \int _1^{K[2]}\frac {a^2-2 K[1]^2 a-2 b K[1]^2}{a K[1]^3+b K[1]^3+a^2 K[1]}dK[1]\right ) \left (a^2+K[2]^2 a+b K[2]^2\right )}{K[2]}dK[2]}\right )+\sqrt {2} a^3 \exp \left (\int _1^x\frac {a^2-2 K[1]^2 a-2 b K[1]^2}{a K[1]^3+b K[1]^3+a^2 K[1]}dK[1]\right )}{x \left (\sqrt {2} a x^2 \exp \left (\int _1^x\frac {a^2-2 K[1]^2 a-2 b K[1]^2}{a K[1]^3+b K[1]^3+a^2 K[1]}dK[1]\right )+\sqrt {2} b x^2 \exp \left (\int _1^x\frac {a^2-2 K[1]^2 a-2 b K[1]^2}{a K[1]^3+b K[1]^3+a^2 K[1]}dK[1]\right )+\sqrt {2} a^2 \exp \left (\int _1^x\frac {a^2-2 K[1]^2 a-2 b K[1]^2}{a K[1]^3+b K[1]^3+a^2 K[1]}dK[1]\right )+2 a \sqrt {-\int _1^x-\frac {\exp \left (2 \int _1^{K[2]}\frac {a^2-2 K[1]^2 a-2 b K[1]^2}{a K[1]^3+b K[1]^3+a^2 K[1]}dK[1]\right ) \left (a^2+K[2]^2 a+b K[2]^2\right )}{K[2]}dK[2]}\right )} \\ y(x)\to -\frac {2 \sqrt {2} a^2 x^2 \exp \left (\int _1^x\frac {a^2-2 K[1]^2 a-2 b K[1]^2}{a K[1]^3+b K[1]^3+a^2 K[1]}dK[1]\right )+\sqrt {2} a x^2 \left (b+x^2\right ) \exp \left (\int _1^x\frac {a^2-2 K[1]^2 a-2 b K[1]^2}{a K[1]^3+b K[1]^3+a^2 K[1]}dK[1]\right )+b x^2 \left (\sqrt {2} x^2 \exp \left (\int _1^x\frac {a^2-2 K[1]^2 a-2 b K[1]^2}{a K[1]^3+b K[1]^3+a^2 K[1]}dK[1]\right )+2 \sqrt {-\int _1^x-\frac {\exp \left (2 \int _1^{K[2]}\frac {a^2-2 K[1]^2 a-2 b K[1]^2}{a K[1]^3+b K[1]^3+a^2 K[1]}dK[1]\right ) \left (a^2+K[2]^2 a+b K[2]^2\right )}{K[2]}dK[2]}\right )+\sqrt {2} a^3 \exp \left (\int _1^x\frac {a^2-2 K[1]^2 a-2 b K[1]^2}{a K[1]^3+b K[1]^3+a^2 K[1]}dK[1]\right )}{x \left (\sqrt {2} a x^2 \exp \left (\int _1^x\frac {a^2-2 K[1]^2 a-2 b K[1]^2}{a K[1]^3+b K[1]^3+a^2 K[1]}dK[1]\right )+\sqrt {2} b x^2 \exp \left (\int _1^x\frac {a^2-2 K[1]^2 a-2 b K[1]^2}{a K[1]^3+b K[1]^3+a^2 K[1]}dK[1]\right )+\sqrt {2} a^2 \exp \left (\int _1^x\frac {a^2-2 K[1]^2 a-2 b K[1]^2}{a K[1]^3+b K[1]^3+a^2 K[1]}dK[1]\right )-2 a \sqrt {-\int _1^x-\frac {\exp \left (2 \int _1^{K[2]}\frac {a^2-2 K[1]^2 a-2 b K[1]^2}{a K[1]^3+b K[1]^3+a^2 K[1]}dK[1]\right ) \left (a^2+K[2]^2 a+b K[2]^2\right )}{K[2]}dK[2]}\right )} \\ \end{align*}