85.67.1 problem 1

Internal problem ID [22892]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 4. Linear differential equations. A Exercises at page 216
Problem number : 1
Date solved : Thursday, October 02, 2025 at 09:16:20 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+3 y&=x^{2}+1 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.036 (sec). Leaf size: 30

ode:=diff(diff(y(x),x),x)+3*y(x) = x^2+1; 
ic:=[y(0) = 0, D(y)(0) = 2]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {2 \sqrt {3}\, \sin \left (\sqrt {3}\, x \right )}{3}-\frac {\cos \left (\sqrt {3}\, x \right )}{9}+\frac {x^{2}}{3}+\frac {1}{9} \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 41

ode=D[y[x],{x,2}]+3*y[x]==x^2+1; 
ic={y[0]==0,Derivative[1][y][0] ==2}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{9} \left (3 x^2+6 \sqrt {3} \sin \left (\sqrt {3} x\right )-\cos \left (\sqrt {3} x\right )+1\right ) \end{align*}

✓ Sympy. Time used: 0.068 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**2 + 3*y(x) + Derivative(y(x), (x, 2)) - 1,0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 2} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {x^{2}}{3} + \frac {2 \sqrt {3} \sin {\left (\sqrt {3} x \right )}}{3} - \frac {\cos {\left (\sqrt {3} x \right )}}{9} + \frac {1}{9} \]