34.1.6 problem 9

Internal problem ID [6030]
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
Date solved : Monday, January 27, 2025 at 01:33:41 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} \left (1+{y^{\prime }}^{2}\right )^{3}&=a^{2} {y^{\prime \prime }}^{2} \end{align*}

✓ Solution by Maple

Time used: 0.047 (sec). Leaf size: 91

dsolve((diff(y(x),x)^2+1)^3=a^2*(diff(y(x),x$2))^2,y(x), singsol=all)
 

\begin{align*} y &= -i x +c_1 \\ y &= i x +c_1 \\ y &= \frac {\left (a +x +c_1 \right ) \left (-a +x +c_1 \right )}{\sqrt {-c_1^{2}-2 c_1 x +a^{2}-x^{2}}}+c_2 \\ y &= \frac {\left (a +x +c_1 \right ) \left (a -x -c_1 \right )}{\sqrt {-c_1^{2}-2 c_1 x +a^{2}-x^{2}}}+c_2 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.661 (sec). Leaf size: 141

DSolve[(D[y[x],x]^2+1)^3==a^2*(D[y[x],{x,2}])^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*}