64.11.37 problem 37

Internal problem ID [13408]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 4, Section 4.3. The method of undetermined coefficients. Exercises page 151
Problem number : 37
Date solved : Wednesday, March 05, 2025 at 09:52:33 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

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

✓ Maple. Time used: 0.019 (sec). Leaf size: 23

ode:=diff(diff(y(x),x),x)+y(x) = 3*x^2-4*sin(x); 
ic:=y(0) = 0, D(y)(0) = 1; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \left (2 x +6\right ) \cos \left (x \right )+3 x^{2}-\sin \left (x \right )-6 \]

✓ Mathematica. Time used: 0.206 (sec). Leaf size: 112

ode=D[y[x],{x,2}]+y[x]==3*x^2-4*Sin[x]; 
ic={y[0]==0,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to -\cos (x) \int _1^0\sin (K[1]) \left (4 \sin (K[1])-3 K[1]^2\right )dK[1]+\cos (x) \int _1^x\sin (K[1]) \left (4 \sin (K[1])-3 K[1]^2\right )dK[1]+\sin (x) \left (\int _1^x\cos (K[2]) \left (3 K[2]^2-4 \sin (K[2])\right )dK[2]-\int _1^0\cos (K[2]) \left (3 K[2]^2-4 \sin (K[2])\right )dK[2]+1\right ) \]

✓ Sympy. Time used: 0.112 (sec). Leaf size: 20

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-3*x**2 + y(x) + 4*sin(x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = 3 x^{2} + \left (2 x + 6\right ) \cos {\left (x \right )} - \sin {\left (x \right )} - 6 \]