Internal problem ID [1588]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 9 Introduction to Linear Higher Order Equations. Section 9.4. Variation of
Parameters for Higher Order Equations. Page 503
Problem number: section 9.4, problem 41.
ODE order: 4.
ODE degree: 1.
CAS Maple gives this as type [[_high_order, _exact, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {x^{4} y^{\prime \prime \prime \prime }+6 x^{3} y^{\prime \prime \prime }+2 x^{2} y^{\prime \prime }-4 y^{\prime } x +4 y-F \relax (x )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 43
dsolve(x^4*diff(y(x),x$4)+6*x^3*diff(y(x),x$3)+2*x^2*diff(y(x),x$2)-4*x*diff(y(x),x)+4*y(x)=F(x),y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{4}+\int \left (2 c_{2} x +c_{3}-\left (\int \left (\int -\frac {c_{1}-\left (\int -F \relax (x )d x \right )}{x^{4}}d x \right )d x \right )\right ) x^{2}d x}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.039 (sec). Leaf size: 102
DSolve[x^4*y''''[x]+6*x^3*y'''[x]+2*x^2*y''[x]-4*x*y'[x]+4*y[x]==f[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x^3 \left (\int _1^x-\frac {f(K[3])}{6 K[3]^2}dK[3]+x \int _1^x\frac {f(K[4])}{12 K[4]^3}dK[4]\right )+x \int _1^x\frac {1}{6} f(K[2])dK[2]+\int _1^x-\frac {1}{12} f(K[1]) K[1]dK[1]+x^3 (c_4 x+c_3)+c_2 x+c_1}{x^2} \\ \end{align*}