40.14.11 problem 32

Internal problem ID [6782]
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 : 32
Date solved : Sunday, March 30, 2025 at 11:22:36 AM
CAS classification : [[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.026 (sec). Leaf size: 49

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

\begin{align*} y &= -\frac {x}{2}-\frac {\sqrt {-4 c_1 x +5 x^{2}+4 c_2}}{2} \\ y &= -\frac {x}{2}+\frac {\sqrt {-4 c_1 x +5 x^{2}+4 c_2}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.645 (sec). Leaf size: 77

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

\begin{align*} y(x)\to -\frac {1}{2} x \left (1+\sqrt {\frac {1}{x^2}} \sqrt {5 x^2+4 c_2 x+4 c_1}\right ) \\ y(x)\to \frac {1}{2} x \left (-1+\sqrt {\frac {1}{x^2}} \sqrt {5 x^2+4 c_2 x+4 c_1}\right ) \\ \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -sqrt(-2*x*Derivative(y(x), (x, 2)) - 4*y(x)*Derivative(y(x), (x, 2)) + 5)/2 + Derivative(y(x), x) + 1/2 cannot be solved by the factorable group method