67.2.49 problem Problem 19(b)

Internal problem ID [13935]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 4, Second and Higher Order Linear Differential Equations. Problems page 221
Problem number : Problem 19(b)
Date solved : Monday, March 31, 2025 at 08:18:40 AM
CAS classification : [[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.051 (sec). Leaf size: 26

ode:=x*diff(diff(y(x),x),x)/(1+y(x))+(y(x)*diff(y(x),x)-x*diff(y(x),x)^2+diff(y(x),x))/(1+y(x))^2 = x*sin(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \,x^{-c_2} {\mathrm e}^{\operatorname {Si}\left (x \right )-\sin \left (x \right )-\frac {\pi \,\operatorname {csgn}\left (x \right )}{2}}-1 \]

✓ Mathematica. Time used: 60.33 (sec). Leaf size: 450

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

\[ y(x)\to -\frac {x \log (x) \exp \left (c_2-\int _1^x\frac {c_1{}^2+\log (K[3]) c_1+c_1+\int _1^{K[3]}-\frac {K[1] \log (K[1]) \sin (K[1])+K[1] \log (K[1]) y(K[1]) \sin (K[1])+y''(K[1])}{y(K[1])+1}dK[1]{}^2+K[3]^2 \log ^2(K[3]) \sin (K[3])+(2 c_1+\log (K[3])+1) \int _1^{K[3]}-\frac {K[1] \log (K[1]) \sin (K[1])+K[1] \log (K[1]) y(K[1]) \sin (K[1])+y''(K[1])}{y(K[1])+1}dK[1]+y(K[3]) \left (K[3]^2 \sin (K[3]) \log ^2(K[3])+\int _1^{K[3]}-\frac {K[1] \log (K[1]) \sin (K[1])+K[1] \log (K[1]) y(K[1]) \sin (K[1])+y''(K[1])}{y(K[1])+1}dK[1]{}^2+c_1 (c_1+\log (K[3])+1)+(2 c_1+\log (K[3])+1) \int _1^{K[3]}-\frac {K[1] \log (K[1]) \sin (K[1])+K[1] \log (K[1]) y(K[1]) \sin (K[1])+y''(K[1])}{y(K[1])+1}dK[1]\right )+K[3] \log (K[3]) y''(K[3])}{K[3] \log (K[3]) (y(K[3])+1) \left (c_1+\int _1^{K[3]}-\frac {K[1] \log (K[1]) \sin (K[1])+K[1] \log (K[1]) y(K[1]) \sin (K[1])+y''(K[1])}{y(K[1])+1}dK[1]\right )}dK[3]\right )+\int _1^x-\frac {K[1] \log (K[1]) \sin (K[1])+K[1] \log (K[1]) y(K[1]) \sin (K[1])+y''(K[1])}{y(K[1])+1}dK[1]+c_1}{\int _1^x-\frac {K[1] \log (K[1]) \sin (K[1])+K[1] \log (K[1]) y(K[1]) \sin (K[1])+y''(K[1])}{y(K[1])+1}dK[1]+c_1} \]

✗ Sympy

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

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