73.16.22 problem 24.4 (d)

Internal problem ID [15463]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 24. Variation of parameters. Additional exercises page 444
Problem number : 24.4 (d)
Date solved : Monday, March 31, 2025 at 01:38:36 PM
CAS classification : [[_high_order, _exact, _linear, _nonhomogeneous]]

\begin{align*} x^{4} y^{\prime \prime \prime \prime }+6 x^{3} y^{\prime \prime \prime }-3 x^{2} y^{\prime \prime }-9 x y^{\prime }+9 y&=12 x \sin \left (x^{2}\right ) \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 69

ode:=x^4*diff(diff(diff(diff(y(x),x),x),x),x)+6*x^3*diff(diff(diff(y(x),x),x),x)-3*x^2*diff(diff(y(x),x),x)-9*x*diff(y(x),x)+9*y(x) = 12*x*sin(x^2); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (-2 x^{4}-2\right ) \sin \left (x^{2}\right )+16 c_4 \,x^{6}+2 \,\operatorname {Ci}\left (x^{2}\right ) x^{6}+16 c_2 \,x^{4}-6 \,\operatorname {Si}\left (x^{2}\right ) x^{4}+c_1 \,x^{2}-4 x^{2} \cos \left (x^{2}\right )+16 c_3}{16 x^{3}} \]

✓ Mathematica. Time used: 0.03 (sec). Leaf size: 101

ode=x^4*D[y[x],{x,4}]+6*x^3*D[y[x],{x,3}]-3*x^2*D[y[x],{x,2}]-9*x*D[y[x],x]+9*y[x]==12*x*Sin[x^2]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {8 x^2 \int _1^x\frac {3}{4} K[1] \sin \left (K[1]^2\right )dK[1]+x^6 \operatorname {CosIntegral}\left (x^2\right )-3 x^4 \text {Si}\left (x^2\right )+8 c_4 x^6+8 c_3 x^4-\sin \left (x^2\right )+x^2 \cos \left (x^2\right )+8 c_2 x^2-x^4 \sin \left (x^2\right )+8 c_1}{8 x^3} \]

✓ Sympy. Time used: 2.353 (sec). Leaf size: 87

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**4*Derivative(y(x), (x, 4)) + 6*x**3*Derivative(y(x), (x, 3)) - 3*x**2*Derivative(y(x), (x, 2)) - 12*x*sin(x**2) - 9*x*Derivative(y(x), x) + 9*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {C_{1}}{x^{3}} + \frac {C_{2}}{x} + C_{3} x + C_{4} x^{3} - \frac {x^{3} \log {\left (x^{2} \right )}}{8} + \frac {x^{3} \log {\left (x^{4} \right )}}{16} + \frac {x^{3} \operatorname {Ci}{\left (x^{2} \right )}}{8} - \frac {x \sin {\left (x^{2} \right )}}{8} - \frac {3 x \operatorname {Si}{\left (x^{2} \right )}}{8} - \frac {\cos {\left (x^{2} \right )}}{4 x} - \frac {\sin {\left (x^{2} \right )}}{8 x^{3}} \]