Internal problem ID [4686]
Book: A treatise on ordinary and partial differential equations by William Woolsey Johnson.
1913
Section: Chapter 1, Nature and meaning of a differential equation between two variables. page
12
Problem number: 9.
ODE order: 2.
ODE degree: 2.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
\[ \boxed {\left (1+{y^{\prime }}^{2}\right )^{3}-a^{2} {y^{\prime \prime }}^{2}=0} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 93
dsolve((diff(y(x),x)^2+1)^3=a^2*(diff(y(x),x$2))^2,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -i x +c_{1} \\ y \left (x \right ) &= i x +c_{1} \\ y \left (x \right ) &= \frac {\left (a +x +c_{1} \right ) \left (-a +x +c_{1} \right )}{\sqrt {a^{2}-c_{1}^{2}-2 c_{1} x -x^{2}}}+c_{2} \\ y \left (x \right ) &= \frac {\left (a +x +c_{1} \right ) \left (a -x -c_{1} \right )}{\sqrt {a^{2}-c_{1}^{2}-2 c_{1} x -x^{2}}}+c_{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.658 (sec). Leaf size: 141
DSolve[(y'[x]^2+1)^3==a^2*(y''[x])^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2-i \sqrt {a^2 \left (-1+c_1{}^2\right )-2 a c_1 x+x^2} \\ y(x)\to i \sqrt {a^2 \left (-1+c_1{}^2\right )-2 a c_1 x+x^2}+c_2 \\ y(x)\to c_2-i \sqrt {a^2 \left (-1+c_1{}^2\right )+2 a c_1 x+x^2} \\ y(x)\to i \sqrt {a^2 \left (-1+c_1{}^2\right )+2 a c_1 x+x^2}+c_2 \\ \end{align*}