Internal problem ID [6860]
Book: Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam
Publishing Co. NY. 6th edition. 1981.
Section: CHAPTER 16. Nonlinear equations. Section 101. Independent variable missing.
EXERCISES Page 324
Problem number: 43.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\[ \boxed {4 y {y^{\prime }}^{2} y^{\prime \prime }-{y^{\prime }}^{4}=3} \]
✓ Solution by Maple
Time used: 0.079 (sec). Leaf size: 111
dsolve(4*y(x)*diff(y(x),x)^2*diff(y(x),x$2)=diff(y(x),x)^4+3,y(x), singsol=all)
\begin{align*} \frac {-4 \left (c_{1} y \left (x \right )-3\right )^{\frac {3}{4}}+\left (-3 x -3 c_{2} \right ) c_{1}}{3 c_{1}} &= 0 \\ \frac {4 \left (c_{1} y \left (x \right )-3\right )^{\frac {3}{4}}+\left (-3 x -3 c_{2} \right ) c_{1}}{3 c_{1}} &= 0 \\ \frac {-4 i \left (c_{1} y \left (x \right )-3\right )^{\frac {3}{4}}+\left (-3 x -3 c_{2} \right ) c_{1}}{3 c_{1}} &= 0 \\ \frac {4 i \left (c_{1} y \left (x \right )-3\right )^{\frac {3}{4}}+\left (-3 x -3 c_{2} \right ) c_{1}}{3 c_{1}} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.242 (sec). Leaf size: 156
DSolve[4*y[x]*(y'[x])^2*y''[x]==(y'[x])^4+3,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {3}{8} e^{-4 c_1} \left (8+\sqrt [3]{6} \left (-e^{4 c_1} (x+c_2)\right ){}^{4/3}\right ) \\ y(x)\to \frac {3}{8} e^{-4 c_1} \left (8+\sqrt [3]{6} \left (-i e^{4 c_1} (x+c_2)\right ){}^{4/3}\right ) \\ y(x)\to \frac {3}{8} e^{-4 c_1} \left (8+\sqrt [3]{6} \left (i e^{4 c_1} (x+c_2)\right ){}^{4/3}\right ) \\ y(x)\to \frac {3}{8} e^{-4 c_1} \left (8+\sqrt [3]{6} \left (e^{4 c_1} (x+c_2)\right ){}^{4/3}\right ) \\ \end{align*}