75.14.8 problem 334

Internal problem ID [16843]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 14. Differential equations admitting of depression of their order. Exercises page 107
Problem number : 334
Date solved : Thursday, March 13, 2025 at 08:53:06 AM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 12

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

\[ y = c_{1} +{\mathrm e}^{x^{2}} c_{2} \]

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 19

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

\[ y(x)\to \frac {c_1 e^{x^2}}{2}+c_2 \]

✓ Sympy. Time used: 0.209 (sec). Leaf size: 53

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

\[ y{\left (x \right )} = C_{1} + x^{2 \left (\operatorname {re}{\left (x\right )}\right )^{2} - 2 \left (\operatorname {im}{\left (x\right )}\right )^{2} + 2} \left (C_{2} \sin {\left (4 \log {\left (x \right )} \left |{\operatorname {re}{\left (x\right )} \operatorname {im}{\left (x\right )}}\right | \right )} + C_{3} \cos {\left (4 \log {\left (x \right )} \operatorname {re}{\left (x\right )} \operatorname {im}{\left (x\right )} \right )}\right ) \]