Internal problem ID [5330]
Book: Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres.
McGraw Hill 1952
Section: Chapter 9. Equations of first order and higher degree. Supplemetary problems. Page
65
Problem number: 24.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _rational]
\[ \boxed {16 y^{3} {y^{\prime }}^{2}-4 x y^{\prime }+y=0} \]
✓ Solution by Maple
Time used: 0.203 (sec). Leaf size: 105
dsolve(16*y(x)^3*diff(y(x),x)^2-4*x*diff(y(x),x)+y(x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {\sqrt {2}\, \sqrt {-x}}{2} \\ y \left (x \right ) &= \frac {\sqrt {2}\, \sqrt {-x}}{2} \\ y \left (x \right ) &= -\frac {\sqrt {x}\, \sqrt {2}}{2} \\ y \left (x \right ) &= \frac {\sqrt {x}\, \sqrt {2}}{2} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \operatorname {RootOf}\left (-\ln \left (x \right )+2 \left (\int _{}^{\textit {\_Z}}-\frac {4 \textit {\_a}^{4}-\sqrt {-4 \textit {\_a}^{4}+1}-1}{\textit {\_a} \left (4 \textit {\_a}^{4}-1\right )}d \textit {\_a} \right )+c_{1} \right ) \sqrt {x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.563 (sec). Leaf size: 303
DSolve[16*y[x]^3*y'[x]^2-4*x*y'[x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -e^{\frac {c_1}{4}} \sqrt [4]{e^{c_1}-i x} \\ y(x)\to -i e^{\frac {c_1}{4}} \sqrt [4]{e^{c_1}-i x} \\ y(x)\to i e^{\frac {c_1}{4}} \sqrt [4]{e^{c_1}-i x} \\ y(x)\to e^{\frac {c_1}{4}} \sqrt [4]{e^{c_1}-i x} \\ y(x)\to -e^{\frac {c_1}{4}} \sqrt [4]{i x+e^{c_1}} \\ y(x)\to -i e^{\frac {c_1}{4}} \sqrt [4]{i x+e^{c_1}} \\ y(x)\to i e^{\frac {c_1}{4}} \sqrt [4]{i x+e^{c_1}} \\ y(x)\to e^{\frac {c_1}{4}} \sqrt [4]{i x+e^{c_1}} \\ y(x)\to 0 \\ y(x)\to -\frac {\sqrt {x}}{\sqrt {2}} \\ y(x)\to -\frac {i \sqrt {x}}{\sqrt {2}} \\ y(x)\to \frac {i \sqrt {x}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {x}}{\sqrt {2}} \\ \end{align*}