77.1.80 problem 106 (page 162)

Internal problem ID [17970]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 106 (page 162)
Date solved : Tuesday, January 28, 2025 at 11:19:03 AM
CAS classification : [[_3rd_order, _quadrature]]

\begin{align*} {y^{\prime \prime \prime }}^{2}+x^{2}&=1 \end{align*}

✓ Solution by Maple

Time used: 0.091 (sec). Leaf size: 91

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

\begin{align*} y &= \frac {\left (2 x^{3}+13 x \right ) \sqrt {-x^{2}+1}}{48}+\frac {\left (4 x^{2}+1\right ) \arcsin \left (x \right )}{16}+\frac {c_{1} x^{2}}{2}+c_{2} x +c_{3} \\ y &= \frac {\left (-2 x^{3}-13 x \right ) \sqrt {-x^{2}+1}}{48}+\frac {\left (-4 x^{2}-1\right ) \arcsin \left (x \right )}{16}+\frac {c_{1} x^{2}}{2}+c_{2} x +c_{3} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.114 (sec). Leaf size: 147

DSolve[D[y[x],{x,3}]^2+x^2==1,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {1}{48} \left (6 \left (2 x^2-1\right ) \arcsin (x)-18 \arctan \left (\frac {\sqrt {1-x^2}}{x+1}\right )+x \sqrt {1-x^2} \left (2 x^2+13\right )\right )+c_3 x^2+c_2 x+c_1 \\ y(x)\to \frac {1}{48} \left (\left (6-12 x^2\right ) \arcsin (x)+18 \arctan \left (\frac {\sqrt {1-x^2}}{x+1}\right )-x \sqrt {1-x^2} \left (2 x^2+13\right )\right )+c_3 x^2+c_2 x+c_1 \\ \end{align*}