54.3.7 problem 1007

Internal problem ID [12302]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1007
Date solved : Wednesday, October 01, 2025 at 01:42:56 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-2 y-4 x^{2} {\mathrm e}^{x^{2}}&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 26

ode:=diff(diff(y(x),x),x)-2*y(x)-4*x^2*exp(x^2) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{\sqrt {2}\, x} c_2 +{\mathrm e}^{-\sqrt {2}\, x} c_1 +{\mathrm e}^{x^{2}} \]

✓ Mathematica. Time used: 0.168 (sec). Leaf size: 106

ode=-4*E^x^2*x^2 - 2*y[x] + D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{-\sqrt {2} x} \left (e^{2 \sqrt {2} x} \int _1^x\sqrt {2} e^{K[1] \left (K[1]-\sqrt {2}\right )} K[1]^2dK[1]+\int _1^x-\sqrt {2} e^{K[2] \left (K[2]+\sqrt {2}\right )} K[2]^2dK[2]+c_1 e^{2 \sqrt {2} x}+c_2\right ) \end{align*}

✓ Sympy. Time used: 0.217 (sec). Leaf size: 27

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

\[ y{\left (x \right )} = C_{1} e^{- \sqrt {2} x} + C_{2} e^{\sqrt {2} x} + e^{x^{2}} \]