40.14.12 problem 33

Internal problem ID [6783]
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 : 33
Date solved : Sunday, March 30, 2025 at 11:22:38 AM
CAS classification : [[_3rd_order, _exact, _nonlinear]]

\begin{align*} \left (1+2 y+3 y^{2}\right ) y^{\prime \prime \prime }+6 y^{\prime } \left (y^{\prime \prime }+{y^{\prime }}^{2}+3 y y^{\prime \prime }\right )&=x \end{align*}

✓ Maple. Time used: 0.034 (sec). Leaf size: 1106

ode:=(1+2*y(x)+3*y(x)^2)*diff(diff(diff(y(x),x),x),x)+6*diff(y(x),x)*(diff(diff(y(x),x),x)+diff(y(x),x)^2+3*y(x)*diff(diff(y(x),x),x)) = x; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Mathematica. Time used: 0.349 (sec). Leaf size: 523

ode=(1+2*y[x]+3*y[x]^2)*D[y[x],{x,3}]+6*D[y[x],x]*( D[y[x],{x,2}]+D[y[x],x]^2+3*y[x]*D[y[x],{x,2}] )==x; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \frac {2^{2/3} \left (9 x^4+108 c_1 x^2+\sqrt {2048+\left (9 x^4+108 c_1 x^2+27 c_3 x+56+216 c_2\right ){}^2}+27 c_3 x+56+216 c_2\right ){}^{2/3}-4 \sqrt [3]{9 x^4+108 c_1 x^2+\sqrt {2048+\left (9 x^4+108 c_1 x^2+27 c_3 x+56+216 c_2\right ){}^2}+27 c_3 x+56+216 c_2}-16 \sqrt [3]{2}}{12 \sqrt [3]{9 x^4+108 c_1 x^2+\sqrt {2048+\left (9 x^4+108 c_1 x^2+27 c_3 x+56+216 c_2\right ){}^2}+27 c_3 x+56+216 c_2}} \\ y(x)\to \frac {1}{24} \left (i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{9 x^4+108 c_1 x^2+\sqrt {2048+\left (9 x^4+108 c_1 x^2+27 c_3 x+56+216 c_2\right ){}^2}+27 c_3 x+56+216 c_2}+\frac {16 \sqrt [3]{2} \left (1+i \sqrt {3}\right )}{\sqrt [3]{9 x^4+108 c_1 x^2+\sqrt {2048+\left (9 x^4+108 c_1 x^2+27 c_3 x+56+216 c_2\right ){}^2}+27 c_3 x+56+216 c_2}}-8\right ) \\ y(x)\to \frac {1}{24} \left (-2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{9 x^4+108 c_1 x^2+\sqrt {2048+\left (9 x^4+108 c_1 x^2+27 c_3 x+56+216 c_2\right ){}^2}+27 c_3 x+56+216 c_2}+\frac {16 \sqrt [3]{2} \left (1-i \sqrt {3}\right )}{\sqrt [3]{9 x^4+108 c_1 x^2+\sqrt {2048+\left (9 x^4+108 c_1 x^2+27 c_3 x+56+216 c_2\right ){}^2}+27 c_3 x+56+216 c_2}}-8\right ) \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x + (18*y(x)*Derivative(y(x), (x, 2)) + 6*Derivative(y(x), x)**2 + 6*Derivative(y(x), (x, 2)))*Derivative(y(x), x) + (3*y(x)**2 + 2*y(x) + 1)*Derivative(y(x), (x, 3)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE (-9*y(x)*Derivative(y(x), (x, 2)) - 3*Derivative(y(x), (x, 2)))/(3*(-1/2 + sqrt(3)*I/2)*(-9*x/4 + sqrt(-4*(-9*y(x)*Derivative(y(x), (x, 2)) - 3*Derivative(y(x), (x, 2)))**3 + (-9*x/2 + 27*y(x)**2*Derivative(y(x), (x, 3))/2 + 9*y(x)*Derivative(y(x), (x, 3)) + 9*Derivative(y(x), (x, 3))/2)**2)/2 + 27*y(x)**2*Derivative(y(x), (x, 3))/4 + 9*y(x)*Derivative(y(x), (x, 3))/2 + 9*Derivative(y(x), (x, 3))/4)**(1/3)) + (-1/2 + sqrt(3)*I/2)*(-9*x/4 + sqrt(-4*(-9*y(x)*Derivative(y(x), (x, 2)) - 3*Derivative(y(x), (x, 2)))**3 + (-9*x/2 + 27*y(x)**2*Derivative(y(x), (x, 3))/2 + 9*y(x)*Derivative(y(x), (x, 3)) + 9*Derivative(y(x), (x, 3))/2)**2)/2 + 27*y(x)**2*Derivative(y(x), (x, 3))/4 + 9*y(x)*Derivative(y(x), (x, 3))/2 + 9*Derivative(y(x), (x, 3))/4)**(1/3)/3 + Derivative(y(x), x) cannot be solved by the factorable group method