Internal problem ID [3338]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 21
Problem number: 596.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_exact, _rational]
Solve \begin {gather*} \boxed {\left (x +y^{2}\right ) y^{\prime }+y-b x -a=0} \end {gather*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 710
dsolve((x+y(x)^2)*diff(y(x),x)+y(x) = b*x+a,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (6 b \,x^{2}+12 a x -12 c_{1}+2 \sqrt {9 b^{2} x^{4}+36 b a \,x^{3}+36 a^{2} x^{2}-36 b c_{1} x^{2}-72 c_{1} a x +16 x^{3}+36 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}-\frac {2 x}{\left (6 b \,x^{2}+12 a x -12 c_{1}+2 \sqrt {9 b^{2} x^{4}+36 b a \,x^{3}+36 a^{2} x^{2}-36 b c_{1} x^{2}-72 c_{1} a x +16 x^{3}+36 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \relax (x ) = -\frac {\left (6 b \,x^{2}+12 a x -12 c_{1}+2 \sqrt {9 b^{2} x^{4}+36 b a \,x^{3}+36 a^{2} x^{2}-36 b c_{1} x^{2}-72 c_{1} a x +16 x^{3}+36 c_{1}^{2}}\right )^{\frac {1}{3}}}{4}+\frac {x}{\left (6 b \,x^{2}+12 a x -12 c_{1}+2 \sqrt {9 b^{2} x^{4}+36 b a \,x^{3}+36 a^{2} x^{2}-36 b c_{1} x^{2}-72 c_{1} a x +16 x^{3}+36 c_{1}^{2}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (6 b \,x^{2}+12 a x -12 c_{1}+2 \sqrt {9 b^{2} x^{4}+36 b a \,x^{3}+36 a^{2} x^{2}-36 b c_{1} x^{2}-72 c_{1} a x +16 x^{3}+36 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {2 x}{\left (6 b \,x^{2}+12 a x -12 c_{1}+2 \sqrt {9 b^{2} x^{4}+36 b a \,x^{3}+36 a^{2} x^{2}-36 b c_{1} x^{2}-72 c_{1} a x +16 x^{3}+36 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (6 b \,x^{2}+12 a x -12 c_{1}+2 \sqrt {9 b^{2} x^{4}+36 b a \,x^{3}+36 a^{2} x^{2}-36 b c_{1} x^{2}-72 c_{1} a x +16 x^{3}+36 c_{1}^{2}}\right )^{\frac {1}{3}}}{4}+\frac {x}{\left (6 b \,x^{2}+12 a x -12 c_{1}+2 \sqrt {9 b^{2} x^{4}+36 b a \,x^{3}+36 a^{2} x^{2}-36 b c_{1} x^{2}-72 c_{1} a x +16 x^{3}+36 c_{1}^{2}}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (6 b \,x^{2}+12 a x -12 c_{1}+2 \sqrt {9 b^{2} x^{4}+36 b a \,x^{3}+36 a^{2} x^{2}-36 b c_{1} x^{2}-72 c_{1} a x +16 x^{3}+36 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {2 x}{\left (6 b \,x^{2}+12 a x -12 c_{1}+2 \sqrt {9 b^{2} x^{4}+36 b a \,x^{3}+36 a^{2} x^{2}-36 b c_{1} x^{2}-72 c_{1} a x +16 x^{3}+36 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 6.43 (sec). Leaf size: 398
DSolve[(x+y[x]^2)y'[x]+y[x]==a+b x,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {-2\ 2^{2/3} x+\sqrt [3]{2} \left (\sqrt {16 x^3+9 \left (2 a x+b x^2+2 c_1\right ){}^2}+6 a x+3 b x^2+6 c_1\right ){}^{2/3}}{2 \sqrt [3]{\sqrt {16 x^3+9 \left (2 a x+b x^2+2 c_1\right ){}^2}+6 a x+3 b x^2+6 c_1}} \\ y(x)\to \frac {4 \sqrt [3]{-2} x+i \left (\sqrt {3}+i\right ) \left (\sqrt {x^2 \left (9 (2 a+b x)^2+16 x\right )+36 c_1 x (2 a+b x)+36 c_1{}^2}+6 a x+3 b x^2+6 c_1\right ){}^{2/3}}{2\ 2^{2/3} \sqrt [3]{\sqrt {16 x^3+9 \left (2 a x+b x^2+2 c_1\right ){}^2}+6 a x+3 b x^2+6 c_1}} \\ y(x)\to \frac {x-i \sqrt {3} x}{\sqrt [3]{2} \sqrt [3]{\sqrt {16 x^3+9 \left (2 a x+b x^2+2 c_1\right ){}^2}+6 a x+3 b x^2+6 c_1}}-\frac {i \left (\sqrt {3}-i\right ) \sqrt [3]{\sqrt {16 x^3+9 \left (2 a x+b x^2+2 c_1\right ){}^2}+6 a x+3 b x^2+6 c_1}}{2\ 2^{2/3}} \\ \end{align*}