10.5.18 problem 25

Internal problem ID [1210]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Section 2.6. Page 100
Problem number : 25
Date solved : Monday, January 27, 2025 at 04:45:19 AM
CAS classification : [[_homogeneous, `class D`], _rational]

\begin{align*} 2 y x +3 x^{2} y+y^{3}+\left (x^{2}+y^{2}\right ) y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.026 (sec). Leaf size: 293

dsolve(2*x*y(x)+3*x^2*y(x)+y(x)^3+(x^2+y(x)^2)*diff(y(x),x) = 0,y(x), singsol=all)
 

\begin{align*} y &= -\frac {2^{{1}/{3}} \left (x^{2} {\mathrm e}^{6 x} c_1^{2}-\frac {2^{{1}/{3}} {\left (\left (1+\sqrt {4 x^{6} {\mathrm e}^{6 x} c_1^{2}+1}\right ) {\mathrm e}^{6 x} c_1^{2}\right )}^{{2}/{3}}}{2}\right ) {\mathrm e}^{-3 x}}{{\left (\left (1+\sqrt {4 x^{6} {\mathrm e}^{6 x} c_1^{2}+1}\right ) {\mathrm e}^{6 x} c_1^{2}\right )}^{{1}/{3}} c_1} \\ y &= -\frac {2^{{1}/{3}} {\mathrm e}^{-3 x} \left (2 x^{2} \left (i \sqrt {3}-1\right ) {\mathrm e}^{6 x} c_1^{2}+2^{{1}/{3}} \left (1+i \sqrt {3}\right ) {\left (\left (1+\sqrt {4 x^{6} {\mathrm e}^{6 x} c_1^{2}+1}\right ) {\mathrm e}^{6 x} c_1^{2}\right )}^{{2}/{3}}\right )}{4 {\left (\left (1+\sqrt {4 x^{6} {\mathrm e}^{6 x} c_1^{2}+1}\right ) {\mathrm e}^{6 x} c_1^{2}\right )}^{{1}/{3}} c_1} \\ y &= \frac {2^{{1}/{3}} {\mathrm e}^{-3 x} \left (2 x^{2} \left (1+i \sqrt {3}\right ) {\mathrm e}^{6 x} c_1^{2}+2^{{1}/{3}} \left (i \sqrt {3}-1\right ) {\left (\left (1+\sqrt {4 x^{6} {\mathrm e}^{6 x} c_1^{2}+1}\right ) {\mathrm e}^{6 x} c_1^{2}\right )}^{{2}/{3}}\right )}{4 {\left (\left (1+\sqrt {4 x^{6} {\mathrm e}^{6 x} c_1^{2}+1}\right ) {\mathrm e}^{6 x} c_1^{2}\right )}^{{1}/{3}} c_1} \\ \end{align*}

✓ Solution by Mathematica

Time used: 60.309 (sec). Leaf size: 383

DSolve[2*x*y[x]+3*x^2*y[x]+y[x]^3+(x^2+y[x]^2)*D[y[x],x] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {e^{-3 x} \left (-2 e^{6 x} x^2+\sqrt [3]{2} \left (\sqrt {4 e^{18 x} x^6+e^{6 (2 x+c_1)}}+e^{6 x+3 c_1}\right ){}^{2/3}\right )}{2^{2/3} \sqrt [3]{\sqrt {4 e^{18 x} x^6+e^{6 (2 x+c_1)}}+e^{6 x+3 c_1}}} \\ y(x)\to \frac {i \left (\sqrt {3}+i\right ) e^{-3 x} \sqrt [3]{\sqrt {4 e^{18 x} x^6+e^{6 (2 x+c_1)}}+e^{6 x+3 c_1}}}{2 \sqrt [3]{2}}+\frac {\left (1+i \sqrt {3}\right ) e^{3 x} x^2}{2^{2/3} \sqrt [3]{\sqrt {4 e^{18 x} x^6+e^{6 (2 x+c_1)}}+e^{6 x+3 c_1}}} \\ y(x)\to \frac {\left (1-i \sqrt {3}\right ) e^{3 x} x^2}{2^{2/3} \sqrt [3]{\sqrt {4 e^{18 x} x^6+e^{6 (2 x+c_1)}}+e^{6 x+3 c_1}}}-\frac {\left (1+i \sqrt {3}\right ) e^{-3 x} \sqrt [3]{\sqrt {4 e^{18 x} x^6+e^{6 (2 x+c_1)}}+e^{6 x+3 c_1}}}{2 \sqrt [3]{2}} \\ \end{align*}