15.19.24 problem 24

Internal problem ID [3308]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 37, page 171
Problem number : 24
Date solved : Monday, January 27, 2025 at 07:33:03 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.033 (sec). Leaf size: 118

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

\begin{align*} x \left (1-\frac {c_{1}}{\left (-2 x +\sqrt {x \left (x +y \left (x \right )\right )}\right ) \left (\frac {-2 x +\sqrt {x \left (x +y \left (x \right )\right )}}{x}\right )^{{1}/{3}} \left (\frac {x +\sqrt {x \left (x +y \left (x \right )\right )}}{x}\right )^{{2}/{3}}}\right ) &= 0 \\ x \left (1+\frac {c_{1}}{\left (2 x +\sqrt {x \left (x +y \left (x \right )\right )}\right ) \left (\frac {-2 x -\sqrt {x \left (x +y \left (x \right )\right )}}{x}\right )^{{1}/{3}} \left (\frac {x -\sqrt {x \left (x +y \left (x \right )\right )}}{x}\right )^{{2}/{3}}}\right ) &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 60.099 (sec). Leaf size: 1178

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

\begin{align*} y(x)\to \frac {\sqrt [3]{2} \sqrt {x} \left (x^{3/2}-2 e^{\frac {3 c_1}{2}}\right )}{\sqrt [3]{-10 e^{\frac {3 c_1}{2}} x^{3/2}+\sqrt {e^{\frac {3 c_1}{2}} \left (4 x^{3/2}+e^{\frac {3 c_1}{2}}\right ){}^3}-2 x^3+e^{3 c_1}}}+\frac {\sqrt [3]{-10 e^{\frac {3 c_1}{2}} x^{3/2}+\sqrt {e^{\frac {3 c_1}{2}} \left (4 x^{3/2}+e^{\frac {3 c_1}{2}}\right ){}^3}-2 x^3+e^{3 c_1}}}{\sqrt [3]{2}}+2 x \\ y(x)\to \frac {\left (1+i \sqrt {3}\right ) \sqrt {x} \left (-x^{3/2}+2 e^{\frac {3 c_1}{2}}\right )}{2^{2/3} \sqrt [3]{-10 e^{\frac {3 c_1}{2}} x^{3/2}+\sqrt {e^{\frac {3 c_1}{2}} \left (4 x^{3/2}+e^{\frac {3 c_1}{2}}\right ){}^3}-2 x^3+e^{3 c_1}}}+\frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{-10 e^{\frac {3 c_1}{2}} x^{3/2}+\sqrt {e^{\frac {3 c_1}{2}} \left (4 x^{3/2}+e^{\frac {3 c_1}{2}}\right ){}^3}-2 x^3+e^{3 c_1}}}{2 \sqrt [3]{2}}+2 x \\ y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt {x} \left (x^{3/2}-2 e^{\frac {3 c_1}{2}}\right )}{2^{2/3} \sqrt [3]{-10 e^{\frac {3 c_1}{2}} x^{3/2}+\sqrt {e^{\frac {3 c_1}{2}} \left (4 x^{3/2}+e^{\frac {3 c_1}{2}}\right ){}^3}-2 x^3+e^{3 c_1}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-10 e^{\frac {3 c_1}{2}} x^{3/2}+\sqrt {e^{\frac {3 c_1}{2}} \left (4 x^{3/2}+e^{\frac {3 c_1}{2}}\right ){}^3}-2 x^3+e^{3 c_1}}}{2 \sqrt [3]{2}}+2 x \\ y(x)\to \frac {\sqrt [3]{2} e^{\frac {3 c_1}{2}} \sqrt {x} \left (2+e^{\frac {3 c_1}{2}} x^{3/2}\right )}{\sqrt [3]{10 e^{\frac {15 c_1}{2}} x^{3/2}+\sqrt {-e^{12 c_1} \left (-1+4 e^{\frac {3 c_1}{2}} x^{3/2}\right ){}^3}-2 e^{9 c_1} x^3+e^{6 c_1}}}+\frac {e^{-3 c_1} \sqrt [3]{10 e^{\frac {15 c_1}{2}} x^{3/2}+\sqrt {-e^{12 c_1} \left (-1+4 e^{\frac {3 c_1}{2}} x^{3/2}\right ){}^3}-2 e^{9 c_1} x^3+e^{6 c_1}}}{\sqrt [3]{2}}+2 x \\ y(x)\to \frac {1}{4} \left (-\frac {2 \sqrt [3]{2} \left (1+i \sqrt {3}\right ) e^{\frac {3 c_1}{2}} \sqrt {x} \left (2+e^{\frac {3 c_1}{2}} x^{3/2}\right )}{\sqrt [3]{10 e^{\frac {15 c_1}{2}} x^{3/2}+\sqrt {-e^{12 c_1} \left (-1+4 e^{\frac {3 c_1}{2}} x^{3/2}\right ){}^3}-2 e^{9 c_1} x^3+e^{6 c_1}}}+i 2^{2/3} \left (\sqrt {3}+i\right ) e^{-3 c_1} \sqrt [3]{10 e^{\frac {15 c_1}{2}} x^{3/2}+\sqrt {-e^{12 c_1} \left (-1+4 e^{\frac {3 c_1}{2}} x^{3/2}\right ){}^3}-2 e^{9 c_1} x^3+e^{6 c_1}}+8 x\right ) \\ y(x)\to \frac {1}{4} \left (\frac {2 i \sqrt [3]{2} \left (\sqrt {3}+i\right ) e^{\frac {3 c_1}{2}} \sqrt {x} \left (2+e^{\frac {3 c_1}{2}} x^{3/2}\right )}{\sqrt [3]{10 e^{\frac {15 c_1}{2}} x^{3/2}+\sqrt {-e^{12 c_1} \left (-1+4 e^{\frac {3 c_1}{2}} x^{3/2}\right ){}^3}-2 e^{9 c_1} x^3+e^{6 c_1}}}-2^{2/3} \left (1+i \sqrt {3}\right ) e^{-3 c_1} \sqrt [3]{10 e^{\frac {15 c_1}{2}} x^{3/2}+\sqrt {-e^{12 c_1} \left (-1+4 e^{\frac {3 c_1}{2}} x^{3/2}\right ){}^3}-2 e^{9 c_1} x^3+e^{6 c_1}}+8 x\right ) \\ \end{align*}