83.30.7 problem 7

Internal problem ID [19315]
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 (D) at page 109
Problem number : 7
Date solved : Monday, March 31, 2025 at 07:06:39 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} \left (x^{2}+1\right ) y^{\prime \prime }+x y^{\prime }+a x&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 14

ode:=(x^2+1)*diff(diff(y(x),x),x)+x*diff(y(x),x)+a*x = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \,\operatorname {arcsinh}\left (x \right )-a x +c_2 \]

✓ Mathematica. Time used: 0.035 (sec). Leaf size: 17

ode=(1+x^2)*D[y[x],{x,2}]+x*D[y[x],x]+a*x==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to -a x+c_1 \text {arcsinh}(x)+c_2 \]

✓ Sympy. Time used: 0.494 (sec). Leaf size: 12

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

\[ y{\left (x \right )} = C_{1} + C_{2} \operatorname {asinh}{\left (x \right )} - a x \]