Internal problem ID [3994]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 26
Problem number: 753.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class G`]]
\[ \boxed {{y^{\prime }}^{2}-8 y=-3 x^{2}} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 153
dsolve(diff(y(x),x)^2+3*x^2 = 8*y(x),y(x), singsol=all)
\begin{align*} y \left (x \right ) = \frac {3 x^{2}}{8}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{6}-18 x \,\textit {\_Z}^{5}+135 x^{2} \textit {\_Z}^{4}-540 x^{3} \textit {\_Z}^{3}+\left (1215 x^{4}-16 c_{1} \right ) \textit {\_Z}^{2}+\left (-1458 x^{5}+32 c_{1} x \right ) \textit {\_Z} +729 x^{6}-16 c_{1} x^{2}\right )^{2}}{8} y \left (x \right ) = \frac {3 x^{2}}{8}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 x \,\textit {\_Z}^{5}+135 x^{2} \textit {\_Z}^{4}+540 x^{3} \textit {\_Z}^{3}+\left (1215 x^{4}-16 c_{1} \right ) \textit {\_Z}^{2}+\left (1458 x^{5}-32 c_{1} x \right ) \textit {\_Z} +729 x^{6}-16 c_{1} x^{2}\right )^{2}}{8} \end{align*}
✓ Solution by Mathematica
Time used: 61.018 (sec). Leaf size: 1865
DSolve[(y'[x])^2+3 x^2==8 y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{96} \left (144 x^2-8\ 2^{2/3} \sqrt [3]{-729 x^4 \cosh (2 c_1)-729 x^4 \sinh (2 c_1)-270 x^2 \cosh (4 c_1)-270 x^2 \sinh (4 c_1)+3 \sqrt {3} \sqrt {x^2 (\cosh (7 c_1)+\sinh (7 c_1)) \left (\left (27 x^2-4\right ) \cosh (c_1)-\left (27 x^2+4\right ) \sinh (c_1)\right ){}^3}+2 \cosh (6 c_1)+2 \sinh (6 c_1)}-\frac {16 \sqrt [3]{2} \left (54 x^2 \cosh (2 c_1)+54 x^2 \sinh (2 c_1)+\cosh (4 c_1)+\sinh (4 c_1)\right )}{\sqrt [3]{-729 x^4 \cosh (2 c_1)-729 x^4 \sinh (2 c_1)-270 x^2 \cosh (4 c_1)-270 x^2 \sinh (4 c_1)+3 \sqrt {3} \sqrt {x^2 (\cosh (7 c_1)+\sinh (7 c_1)) \left (\left (27 x^2-4\right ) \cosh (c_1)-\left (27 x^2+4\right ) \sinh (c_1)\right ){}^3}+2 \cosh (6 c_1)+2 \sinh (6 c_1)}}+32 \cosh (2 c_1)+32 \sinh (2 c_1)\right ) y(x)\to \frac {1}{192} \left (288 x^2+8\ 2^{2/3} \left (1-i \sqrt {3}\right ) \sqrt [3]{-729 x^4 \cosh (2 c_1)-729 x^4 \sinh (2 c_1)-270 x^2 \cosh (4 c_1)-270 x^2 \sinh (4 c_1)+3 \sqrt {3} \sqrt {x^2 (\cosh (7 c_1)+\sinh (7 c_1)) \left (\left (27 x^2-4\right ) \cosh (c_1)-\left (27 x^2+4\right ) \sinh (c_1)\right ){}^3}+2 \cosh (6 c_1)+2 \sinh (6 c_1)}+\frac {16 \sqrt [3]{2} \left (1+i \sqrt {3}\right ) \left (54 x^2 \cosh (2 c_1)+54 x^2 \sinh (2 c_1)+\cosh (4 c_1)+\sinh (4 c_1)\right )}{\sqrt [3]{-729 x^4 \cosh (2 c_1)-729 x^4 \sinh (2 c_1)-270 x^2 \cosh (4 c_1)-270 x^2 \sinh (4 c_1)+3 \sqrt {3} \sqrt {x^2 (\cosh (7 c_1)+\sinh (7 c_1)) \left (\left (27 x^2-4\right ) \cosh (c_1)-\left (27 x^2+4\right ) \sinh (c_1)\right ){}^3}+2 \cosh (6 c_1)+2 \sinh (6 c_1)}}+64 \cosh (2 c_1)+64 \sinh (2 c_1)\right ) y(x)\to \frac {1}{192} \left (288 x^2+8\ 2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{-729 x^4 \cosh (2 c_1)-729 x^4 \sinh (2 c_1)-270 x^2 \cosh (4 c_1)-270 x^2 \sinh (4 c_1)+3 \sqrt {3} \sqrt {x^2 (\cosh (7 c_1)+\sinh (7 c_1)) \left (\left (27 x^2-4\right ) \cosh (c_1)-\left (27 x^2+4\right ) \sinh (c_1)\right ){}^3}+2 \cosh (6 c_1)+2 \sinh (6 c_1)}+\frac {16 \sqrt [3]{2} \left (1-i \sqrt {3}\right ) \left (54 x^2 \cosh (2 c_1)+54 x^2 \sinh (2 c_1)+\cosh (4 c_1)+\sinh (4 c_1)\right )}{\sqrt [3]{-729 x^4 \cosh (2 c_1)-729 x^4 \sinh (2 c_1)-270 x^2 \cosh (4 c_1)-270 x^2 \sinh (4 c_1)+3 \sqrt {3} \sqrt {x^2 (\cosh (7 c_1)+\sinh (7 c_1)) \left (\left (27 x^2-4\right ) \cosh (c_1)-\left (27 x^2+4\right ) \sinh (c_1)\right ){}^3}+2 \cosh (6 c_1)+2 \sinh (6 c_1)}}+64 \cosh (2 c_1)+64 \sinh (2 c_1)\right ) y(x)\to \frac {1}{96} \left (144 x^2-8\ 2^{2/3} \sqrt [3]{729 x^4 \cosh (2 c_1)+729 x^4 \sinh (2 c_1)-270 x^2 \cosh (4 c_1)-270 x^2 \sinh (4 c_1)+3 \sqrt {3} \sqrt {x^2 (\cosh (7 c_1)+\sinh (7 c_1)) \left (\left (27 x^2+4\right ) \cosh (c_1)+\left (4-27 x^2\right ) \sinh (c_1)\right ){}^3}-2 \cosh (6 c_1)-2 \sinh (6 c_1)}+\frac {16 \sqrt [3]{2} \left (54 x^2 \cosh (2 c_1)+54 x^2 \sinh (2 c_1)-\cosh (4 c_1)-\sinh (4 c_1)\right )}{\sqrt [3]{729 x^4 \cosh (2 c_1)+729 x^4 \sinh (2 c_1)-270 x^2 \cosh (4 c_1)-270 x^2 \sinh (4 c_1)+3 \sqrt {3} \sqrt {x^2 (\cosh (7 c_1)+\sinh (7 c_1)) \left (\left (27 x^2+4\right ) \cosh (c_1)+\left (4-27 x^2\right ) \sinh (c_1)\right ){}^3}-2 \cosh (6 c_1)-2 \sinh (6 c_1)}}-32 \cosh (2 c_1)-32 \sinh (2 c_1)\right ) y(x)\to \frac {1}{192} \left (288 x^2+8\ 2^{2/3} \left (1-i \sqrt {3}\right ) \sqrt [3]{729 x^4 \cosh (2 c_1)+729 x^4 \sinh (2 c_1)-270 x^2 \cosh (4 c_1)-270 x^2 \sinh (4 c_1)+3 \sqrt {3} \sqrt {x^2 (\cosh (7 c_1)+\sinh (7 c_1)) \left (\left (27 x^2+4\right ) \cosh (c_1)+\left (4-27 x^2\right ) \sinh (c_1)\right ){}^3}-2 \cosh (6 c_1)-2 \sinh (6 c_1)}-\frac {16 i \sqrt [3]{2} \left (\sqrt {3}-i\right ) \left (54 x^2 \cosh (2 c_1)+54 x^2 \sinh (2 c_1)-\cosh (4 c_1)-\sinh (4 c_1)\right )}{\sqrt [3]{729 x^4 \cosh (2 c_1)+729 x^4 \sinh (2 c_1)-270 x^2 \cosh (4 c_1)-270 x^2 \sinh (4 c_1)+3 \sqrt {3} \sqrt {x^2 (\cosh (7 c_1)+\sinh (7 c_1)) \left (\left (27 x^2+4\right ) \cosh (c_1)+\left (4-27 x^2\right ) \sinh (c_1)\right ){}^3}-2 \cosh (6 c_1)-2 \sinh (6 c_1)}}-64 \cosh (2 c_1)-64 \sinh (2 c_1)\right ) y(x)\to \frac {1}{192} \left (288 x^2+8\ 2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{729 x^4 \cosh (2 c_1)+729 x^4 \sinh (2 c_1)-270 x^2 \cosh (4 c_1)-270 x^2 \sinh (4 c_1)+3 \sqrt {3} \sqrt {x^2 (\cosh (7 c_1)+\sinh (7 c_1)) \left (\left (27 x^2+4\right ) \cosh (c_1)+\left (4-27 x^2\right ) \sinh (c_1)\right ){}^3}-2 \cosh (6 c_1)-2 \sinh (6 c_1)}+\frac {16 i \sqrt [3]{2} \left (\sqrt {3}+i\right ) \left (54 x^2 \cosh (2 c_1)+54 x^2 \sinh (2 c_1)-\cosh (4 c_1)-\sinh (4 c_1)\right )}{\sqrt [3]{729 x^4 \cosh (2 c_1)+729 x^4 \sinh (2 c_1)-270 x^2 \cosh (4 c_1)-270 x^2 \sinh (4 c_1)+3 \sqrt {3} \sqrt {x^2 (\cosh (7 c_1)+\sinh (7 c_1)) \left (\left (27 x^2+4\right ) \cosh (c_1)+\left (4-27 x^2\right ) \sinh (c_1)\right ){}^3}-2 \cosh (6 c_1)-2 \sinh (6 c_1)}}-64 \cosh (2 c_1)-64 \sinh (2 c_1)\right ) \end{align*}