83.27.17 problem 17

Internal problem ID [19292]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VII. Exact differential equations and certain particular forms of equations. Exercise VII (A) at page 104
Problem number : 17
Date solved : Monday, March 31, 2025 at 07:05:51 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{5} y^{\prime \prime }+3 x^{3} y^{\prime }+\left (3-6 x \right ) x^{2} y&=x^{4}+2 x -5 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 111

ode:=x^5*diff(diff(y(x),x),x)+3*x^3*diff(y(x),x)+(3-6*x)*x^2*y(x) = x^4+2*x-5; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {-48 \,\operatorname {Ei}_{1}\left (\frac {3}{x}\right ) x^{5} {\mathrm e}^{\frac {3}{x}}+243 x^{5} {\mathrm e}^{\frac {3}{x}} c_1 +\left (48 x^{5}+144 x^{4}+216 x^{3}+216 x^{2}+162 x \right ) \ln \left (x \right )+405+\left (1944 c_2 +109\right ) x^{5}+\left (5832 c_2 +471\right ) x^{4}+\left (8748 c_2 +774\right ) x^{3}+\left (8748 c_2 +846\right ) x^{2}+\left (6561 c_2 +675\right ) x}{243 x^{2}} \]

✓ Mathematica. Time used: 0.208 (sec). Leaf size: 155

ode=x^5*D[y[x],{x,2}]+3*x^3*D[y[x],x]+(3-6*x)*x^2*y[x]==x^4+2*x-5; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {48 e^{3/x} x^5 \operatorname {ExpIntegralEi}\left (-\frac {3}{x}\right )+109 x^5-48 x^5 \log \left (\frac {3}{x}\right )+243 c_1 e^{3/x} x^5+e^{3/x} x^5 \Gamma \left (5,\frac {3}{x}\right ) \left (2 \log \left (\frac {1}{x}\right )+2 \log (x)+\log (9)+c_2\right )+471 x^4-144 x^4 \log \left (\frac {3}{x}\right )+774 x^3-216 x^3 \log \left (\frac {3}{x}\right )+846 x^2-216 x^2 \log \left (\frac {3}{x}\right )+675 x-162 x \log \left (\frac {3}{x}\right )+405}{243 x^2} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**5*Derivative(y(x), (x, 2)) - x**4 + 3*x**3*Derivative(y(x), x) + x**2*(3 - 6*x)*y(x) - 2*x + 5,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (x**3*(-x**2*Derivative(y(x), (x, 2)) + x + 6*y(x)) - 3*x**2*y(x) + 2*x - 5)/(3*x**3) cannot be solved by the factorable group method