40.14.13 problem 34

Internal problem ID [6784]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 19. Linear equations with variable coefficients (Misc. types). Supplemetary problems. Page 132
Problem number : 34
Date solved : Sunday, March 30, 2025 at 11:22:39 AM
CAS classification : [[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]]

\begin{align*} 3 x \left (y^{2} y^{\prime \prime \prime }+6 y y^{\prime } y^{\prime \prime }+2 {y^{\prime }}^{3}\right )-3 y \left (y y^{\prime \prime }+2 {y^{\prime }}^{2}\right )&=-\frac {2}{x} \end{align*}

✓ Maple. Time used: 0.026 (sec). Leaf size: 102

ode:=3*x*(y(x)^2*diff(diff(diff(y(x),x),x),x)+6*y(x)*diff(y(x),x)*diff(diff(y(x),x),x)+2*diff(y(x),x)^3)-3*y(x)*(y(x)*diff(diff(y(x),x),x)+2*diff(y(x),x)^2) = -2/x; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {\left (8 \ln \left (x \right ) x -8 c_2 \,x^{3}+\left (12 c_1 -4\right ) x +8 c_3 \right )^{{1}/{3}}}{2} \\ y &= -\frac {\left (8 \ln \left (x \right ) x -8 c_2 \,x^{3}+\left (12 c_1 -4\right ) x +8 c_3 \right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{4} \\ y &= \frac {\left (8 \ln \left (x \right ) x -8 c_2 \,x^{3}+\left (12 c_1 -4\right ) x +8 c_3 \right )^{{1}/{3}} \left (-1+i \sqrt {3}\right )}{4} \\ \end{align*}

✓ Mathematica. Time used: 0.509 (sec). Leaf size: 121

ode=3*x*( y[x]^2* D[y[x],{x,3}]+6*y[x]*D[y[x],x]*D[y[x],{x,2}]+2*D[y[x],x]^3   )-3*y[x]* (y[x]*D[y[x],{x,2}]+2* D[y[x],x]^2  )==-2/x; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\sqrt [3]{-\frac {1}{6}} \sqrt [3]{6 c_3 x^3+6 x \log (x)+(3+9 c_2) x+2 c_1} \\ y(x)\to \sqrt [3]{c_3 x^3+x \log (x)+\frac {1}{2} (1+3 c_2) x+\frac {c_1}{3}} \\ y(x)\to (-1)^{2/3} \sqrt [3]{c_3 x^3+x \log (x)+\frac {1}{2} (1+3 c_2) x+\frac {c_1}{3}} \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*x*(y(x)**2*Derivative(y(x), (x, 3)) + 6*y(x)*Derivative(y(x), x)*Derivative(y(x), (x, 2)) + 2*Derivative(y(x), x)**3) - (3*y(x)*Derivative(y(x), (x, 2)) + 6*Derivative(y(x), x)**2)*y(x) + 2/x,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out