71.13.7 problem 7

Internal problem ID [14456]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 5. The Laplace Transform Method. Exercises 5.2, page 248
Problem number : 7
Date solved : Monday, March 31, 2025 at 12:27:34 PM
CAS classification : [[_high_order, _missing_y]]

\begin{align*} y^{\prime \prime \prime \prime }-2 y^{\prime \prime \prime }+y^{\prime \prime }&=x \,{\mathrm e}^{x}-3 x^{2} \end{align*}

Using Laplace method

✓ Maple. Time used: 0.137 (sec). Leaf size: 79

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-2*diff(diff(diff(y(x),x),x),x)+diff(diff(y(x),x),x) = x*exp(x)-3*x^2; 
dsolve(ode,y(x),method='laplace');
 

\[ y = -26-2 x^{3}-9 x^{2}-\frac {x^{4}}{4}+y \left (0\right )+\frac {{\mathrm e}^{x} \left (x^{3}+6 x y^{\prime \prime \prime }\left (0\right )-6 x y^{\prime \prime }\left (0\right )-6 x^{2}-12 y^{\prime \prime \prime }\left (0\right )+18 y^{\prime \prime }\left (0\right )-18 x +156\right )}{6}-y^{\prime \prime }\left (0\right ) \left (3+2 x \right )+y^{\prime \prime \prime }\left (0\right ) \left (2+x \right )+x \left (-23+y^{\prime }\left (0\right )\right ) \]

✓ Mathematica. Time used: 39.373 (sec). Leaf size: 91

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

\[ y(x)\to \int _1^x\int _1^{K[4]}e^{K[3]} \left (c_1+c_2 K[3]+\int _1^{K[3]}K[1]^2 \left (3 e^{-K[1]} K[1]-1\right )dK[1]+K[3] \int _1^{K[3]}\left (K[2]-3 e^{-K[2]} K[2]^2\right )dK[2]\right )dK[3]dK[4]+c_4 x+c_3 \]

✓ Sympy. Time used: 0.200 (sec). Leaf size: 39

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

\[ y{\left (x \right )} = C_{1} + C_{4} e^{x} - \frac {x^{4}}{4} - 2 x^{3} - 9 x^{2} + x \left (C_{2} + \left (C_{3} + \frac {x^{2}}{6} - x\right ) e^{x}\right ) \]