18.2.15 problem Problem 15.33

Internal problem ID [3498]
Book : Mathematical methods for physics and engineering, Riley, Hobson, Bence, second edition, 2002
Section : Chapter 15, Higher order ordinary differential equations. 15.4 Exercises, page 523
Problem number : Problem 15.33
Date solved : Sunday, March 30, 2025 at 01:45:03 AM
CAS classification : [[_3rd_order, _exact, _nonlinear]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 81

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

\begin{align*} y &= -\frac {\sqrt {2}\, {\mathrm e}^{-x} \sqrt {-4 \left (\left (-\frac {\cos \left (x \right )}{4}+\frac {\sin \left (x \right )}{4}+\left (x -1\right ) c_1 +c_3 \right ) {\mathrm e}^{x}-c_2 \right ) {\mathrm e}^{x}}}{2} \\ y &= \frac {\sqrt {2}\, {\mathrm e}^{-x} \sqrt {-4 \left (\left (-\frac {\cos \left (x \right )}{4}+\frac {\sin \left (x \right )}{4}+\left (x -1\right ) c_1 +c_3 \right ) {\mathrm e}^{x}-c_2 \right ) {\mathrm e}^{x}}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.458 (sec). Leaf size: 88

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

\begin{align*} y(x)\to -\frac {\sqrt {-\sin (x)+\cos (x)+2 c_1 x+2 c_3 e^{-x}-2 c_1-4 c_2}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {-\sin (x)+\cos (x)+2 c_1 x+2 c_3 e^{-x}-2 c_1-4 c_2}}{\sqrt {2}} \\ \end{align*}

✗ Sympy

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

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