71.13.7 problem 7

Internal problem ID [14527]
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 : Tuesday, January 28, 2025 at 06:42:59 AM
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

✓ Solution by Maple

Time used: 10.292 (sec). Leaf size: 79

dsolve(diff(y(x),x$4)-2*diff(y(x),x$3)+diff(y(x),x$2)=x*exp(x)-3*x^2,y(x), singsol=all)
 

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

✓ Solution by Mathematica

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

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

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