29.21.20 problem 596
Internal
problem
ID
[5190]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
21
Problem
number
:
596
Date
solved
:
Monday, January 27, 2025 at 10:18:37 AM
CAS
classification
:
[_exact, _rational]
\begin{align*} \left (x +y^{2}\right ) y^{\prime }+y&=b x +a \end{align*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 580
dsolve((x+y(x)^2)*diff(y(x),x)+y(x) = b*x+a,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \frac {\left (6 b \,x^{2}+12 a x -12 c_{1} +2 \sqrt {9 b^{2} x^{4}+36 a \,x^{3} b +36 a^{2} x^{2}-36 b c_{1} x^{2}-72 a x c_{1} +16 x^{3}+36 c_{1}^{2}}\right )^{{2}/{3}}-4 x}{2 \left (6 b \,x^{2}+12 a x -12 c_{1} +2 \sqrt {9 b^{2} x^{4}+36 a \,x^{3} b +36 a^{2} x^{2}-36 b c_{1} x^{2}-72 a x c_{1} +16 x^{3}+36 c_{1}^{2}}\right )^{{1}/{3}}} \\
y \left (x \right ) &= -\frac {i \sqrt {3}\, \left (6 b \,x^{2}+12 a x -12 c_{1} +2 \sqrt {9 b^{2} x^{4}+36 a \,x^{3} b +36 a^{2} x^{2}-36 b c_{1} x^{2}-72 a x c_{1} +16 x^{3}+36 c_{1}^{2}}\right )^{{2}/{3}}+4 i \sqrt {3}\, x +\left (6 b \,x^{2}+12 a x -12 c_{1} +2 \sqrt {9 b^{2} x^{4}+36 a \,x^{3} b +36 a^{2} x^{2}-36 b c_{1} x^{2}-72 a x c_{1} +16 x^{3}+36 c_{1}^{2}}\right )^{{2}/{3}}-4 x}{4 \left (6 b \,x^{2}+12 a x -12 c_{1} +2 \sqrt {9 b^{2} x^{4}+36 a \,x^{3} b +36 a^{2} x^{2}-36 b c_{1} x^{2}-72 a x c_{1} +16 x^{3}+36 c_{1}^{2}}\right )^{{1}/{3}}} \\
y \left (x \right ) &= \frac {i \sqrt {3}\, \left (6 b \,x^{2}+12 a x -12 c_{1} +2 \sqrt {9 b^{2} x^{4}+36 a \,x^{3} b +36 a^{2} x^{2}-36 b c_{1} x^{2}-72 a x c_{1} +16 x^{3}+36 c_{1}^{2}}\right )^{{2}/{3}}+4 i \sqrt {3}\, x -\left (6 b \,x^{2}+12 a x -12 c_{1} +2 \sqrt {9 b^{2} x^{4}+36 a \,x^{3} b +36 a^{2} x^{2}-36 b c_{1} x^{2}-72 a x c_{1} +16 x^{3}+36 c_{1}^{2}}\right )^{{2}/{3}}+4 x}{4 \left (6 b \,x^{2}+12 a x -12 c_{1} +2 \sqrt {9 b^{2} x^{4}+36 a \,x^{3} b +36 a^{2} x^{2}-36 b c_{1} x^{2}-72 a x c_{1} +16 x^{3}+36 c_{1}^{2}}\right )^{{1}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 5.750 (sec). Leaf size: 420
DSolve[(x+y[x]^2)D[y[x],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 {i \sqrt [3]{2} \left (\sqrt {3}+i\right ) \left (\sqrt {36 a^2 x^2+36 a b x^3+72 a c_1 x+9 b^2 x^4+36 b c_1 x^2+16 x^3+36 c_1{}^2}+6 a x+3 b x^2+6 c_1\right ){}^{2/3}+2\ 2^{2/3} \left (1+i \sqrt {3}\right ) x}{4 \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*}