40.14.14 problem 35

Internal problem ID [6785]
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 : 35
Date solved : Sunday, March 30, 2025 at 11:22:40 AM
CAS classification : [[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]]

\begin{align*} y y^{\prime \prime \prime }+3 y^{\prime } y^{\prime \prime }-2 y y^{\prime \prime }-2 {y^{\prime }}^{2}+y y^{\prime }&={\mathrm e}^{2 x} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 51

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

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

✓ Mathematica. Time used: 0.341 (sec). Leaf size: 65

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

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

✗ Sympy

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

NotImplementedError : The given ODE -sqrt(y(x)**2 - 10*y(x)*Derivative(y(x), (x, 2)) + 8*y(x)*Derivative(y(x), (x, 3)) - 8*exp(2*x) + 9*Derivative(y(x), (x, 2))**2)/4 - y(x)/4 + Derivative(y(x), x) - 3*Derivative(y(x), (x, 2))/4 cannot be solved by the factorable group method