2.3.19 Problem 19

Solved as higher order Euler type ode
Maple
Mathematica
Sympy

Internal problem ID [8877]
Book : Own collection of miscellaneous problems
Section : section 3.0
Problem number : 19
Date solved : Sunday, March 30, 2025 at 01:45:25 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

Solved as higher order Euler type ode

Time used: 4.242 (sec)

The ode can be normalized and rewritten as Euler ode.

This is Euler ODE of higher order. Let y=xλ. Hence

y=λxλ1y=λ(λ1)xλ2y=λ(λ1)(λ2)xλ3

Substituting these back into

x4y+x3y+x2y+xy=x

gives

xλxλ1+x2λ(λ1)xλ2+x3λ(λ1)(λ2)xλ3+xλ=0

Which simplifies to

λxλ+λ(λ1)xλ+λ(λ1)(λ2)xλ+xλ=0

And since xλ0 then dividing through by xλ, the above becomes

λ+λ(λ1)+λ(λ1)(λ2)+1=0

Simplifying gives the characteristic equation as

λ32λ2+2λ+1=0

Solving the above gives the following roots

λ1=(188+12249)1/36+43(188+12249)1/3+23λ2=(188+12249)1/31223(188+12249)1/3+23+i3((188+12249)1/3643(188+12249)1/3)2λ3=(188+12249)1/31223(188+12249)1/3+23i3((188+12249)1/3643(188+12249)1/3)2

This table summarises the result

root multiplicity type of root
(188+12249)1/31223(188+12249)1/3+23±3((188+12249)1/3643(188+12249)1/3)2i 1 complex conjugate root
(188+12249)1/36+43(188+12249)1/3+23 1 real root

The solution is generated by going over the above table. For each real root λ of multiplicity one generates a c1xλ basis solution. Each real root of multiplicty two, generates c1xλ and c2xλln(x) basis solutions. Each real root of multiplicty three, generates c1xλ and c2xλln(x) and c3xλln(x)2 basis solutions, and so on. Each complex root α±iβ of multiplicity one generates xα(c1cos(βln(x))+c2sin(βln(x))) basis solutions. And each complex root α±iβ of multiplicity two generates ln(x)xα(c1cos(βln(x))+c2sin(βln(x))) basis solutions. And each complex root α±iβ of multiplicity three generates ln(x)2xα(c1cos(βln(x))+c2sin(βln(x))) basis solutions. And so on. Using the above show that the solution is

y=x(188+12249)1/31223(188+12249)1/3+23(c1cos(3((188+12249)1/3643(188+12249)1/3)ln(x)2)c2sin(3((188+12249)1/3643(188+12249)1/3)ln(x)2))+c3x(188+12249)1/36+43(188+12249)1/3+23

The fundamental set of solutions for the homogeneous solution are the following

y1=x(188+12249)1/31223(188+12249)1/3+23cos(3((188+12249)1/3643(188+12249)1/3)ln(x)2)y2=x(188+12249)1/31223(188+12249)1/3+23sin(3((188+12249)1/3643(188+12249)1/3)ln(x)2)y3=x(188+12249)1/36+43(188+12249)1/3+23

This is higher order nonhomogeneous Euler type ODE. Let the solution be

y=yh+yp

Where yh is the solution to the homogeneous Euler ODE And yp is a particular solution to the nonhomogeneous Euler ODE. yh is the solution to

yx+x2y+x3y+y=0

Now the particular solution to the given ODE is found

yx+x2y+x3y+y=x

Let the particular solution be

yp=U1y1+U2y2+U3y3

Where yi are the basis solutions found above for the homogeneous solution yh and Ui(x) are functions to be determined as follows

Ui=(1)niF(x)Wi(x)aW(x)dx

Where W(x) is the Wronskian and Wi(x) is the Wronskian that results after deleting the last row and the i-th column of the determinant and n is the order of the ODE or equivalently, the number of basis solutions, and a is the coefficient of the leading derivative in the ODE, and F(x) is the RHS of the ODE. Therefore, the first step is to find the Wronskian W(x). This is given by

W(x)=|y1y2y3y1y2y3y1y2y3|

Substituting the fundamental set of solutions yi found above in the Wronskian gives

W=Expression too large to display|W|=Expression too large to display

The determinant simplifies to

|W|=832x

Now we determine Wi for each Ui.

W1(x)=det[x(188+12249)2/3+8(188+12249)1/3812(188+12249)1/3sin(3((188+12383)2/3+8)ln(x)12(188+12383)1/3)x(188+12249)1/36+43(188+12249)1/3+23x(188+12383)2/34(188+12383)1/3812(188+12383)1/3(((188+12383)2/3+8(188+12383)1/38)sin(3((188+12383)2/3+8)ln(x)12(188+12383)1/3)+3((188+12383)2/3+8)cos(3((188+12383)2/3+8)ln(x)12(188+12383)1/3))12(188+12383)1/3x(188+12249)2/3+2(188+12249)1/386(188+12249)1/3((188+12249)2/34(188+12249)1/38)6(188+12249)1/3]=3((2i3239)(188+12249)1/3+i249+47i3473983)x2i3(188+12383)1/3+9i83+47i33383+(188+12383)2/3+2(188+12383)1/3473(188+12383)2/323((2i3+239)(188+12249)1/3+i249+47i3+4739+83)x(2i3+2)(188+12383)1/3+(188+12383)2/3+(9i33)8347i3473(188+12383)2/32(188+12249)2/3
W2(x)=det[x(188+12249)2/3+8(188+12249)1/3812(188+12249)1/3cos(3((188+12383)2/3+8)ln(x)12(188+12383)1/3)x(188+12249)1/36+43(188+12249)1/3+23(((188+12383)2/38(188+12383)1/3+8)cos(3((188+12383)2/3+8)ln(x)12(188+12383)1/3)+3((188+12383)2/3+8)sin(3((188+12383)2/3+8)ln(x)12(188+12383)1/3))x(188+12383)2/34(188+12383)1/3812(188+12383)1/312(188+12383)1/3x(188+12249)2/3+2(188+12249)1/386(188+12249)1/3((188+12249)2/34(188+12249)1/38)6(188+12249)1/3]=((i33)(188+12249)1/3+47i32+9i832+92492+1412)x2i3(188+12383)1/3+9i83+47i33383+(188+12383)2/3+2(188+12383)1/3473(188+12383)2/3x(2i3+2)(188+12383)1/3+(188+12383)2/3+(9i33)8347i3473(188+12383)2/3((i3+3)(188+12249)1/3+47i32+9i832924921412)3(188+12249)2/3
W3(x)=det[x(188+12249)2/3+8(188+12249)1/3812(188+12249)1/3cos(3((188+12383)2/3+8)ln(x)12(188+12383)1/3)x(188+12249)2/3+8(188+12249)1/3812(188+12249)1/3sin(3((188+12383)2/3+8)ln(x)12(188+12383)1/3)(((188+12383)2/38(188+12383)1/3+8)cos(3((188+12383)2/3+8)ln(x)12(188+12383)1/3)+3((188+12383)2/3+8)sin(3((188+12383)2/3+8)ln(x)12(188+12383)1/3))x(188+12383)2/34(188+12383)1/3812(188+12383)1/312(188+12383)1/3x(188+12383)2/34(188+12383)1/3812(188+12383)1/3(((188+12383)2/3+8(188+12383)1/38)sin(3((188+12383)2/3+8)ln(x)12(188+12383)1/3)+3((188+12383)2/3+8)cos(3((188+12383)2/3+8)ln(x)12(188+12383)1/3))12(188+12383)1/3]=3((188+12383)2/3+8)x(188+12383)2/3+2(188+12383)1/386(188+12383)1/312(188+12383)1/3

Now we are ready to evaluate each Ui(x).

U1=(1)31F(x)W1(x)aW(x)dx=(1)2(x)(3((2i3239)(188+12249)1/3+i249+47i3473983)x2i3(188+12383)1/3+9i83+47i33383+(188+12383)2/3+2(188+12383)1/3473(188+12383)2/323((2i3+239)(188+12249)1/3+i249+47i3+4739+83)x(2i3+2)(188+12383)1/3+(188+12383)2/3+(9i33)8347i3473(188+12383)2/32(188+12249)2/3)(x3)(832x)dx=3x(((2i3239)(188+12249)1/3+i249+47i3473983)x2i3(188+12383)1/3+9i83+47i33383+(188+12383)2/3+2(188+12383)1/3473(188+12383)2/3((2i3+239)(188+12249)1/3+i249+47i3+4739+83)x(2i3+2)(188+12383)1/3+(188+12383)2/3+(9i33)8347i3473(188+12383)2/3)2(188+12249)2/3x2832dx=(3(((2i3239)(188+12249)1/3+i249+47i3473983)x2i3(188+12383)1/3+9i83+47i33383+(188+12383)2/3+2(188+12383)1/3473(188+12383)2/3((2i3+239)(188+12249)1/3+i249+47i3+4739+83)x(2i3+2)(188+12383)1/3+(188+12383)2/3+(9i33)8347i3473(188+12383)2/3)8383x(188+12249)2/3)dx=(3i83332+383332i(188+12383)2/3396(188+12383)2/396+3i(188+12383)2/3832656+(188+12383)2/33832656+(188+12383)1/36i312+112)e((2i3+2)(188+12383)1/3+(188+12383)2/3+(9i33)8347i347)ln(x)3(188+12383)2/3+(3i83332+383332+i(188+12383)2/3396(188+12383)2/3963i(188+12383)2/3832656+(188+12383)2/33832656+(188+12383)1/36+i312+112)e(2i3(188+12383)1/3+9i83+47i33383+(188+12383)2/3+2(188+12383)1/347)ln(x)3(188+12383)2/3(188+12383)1/3
U2=(1)32F(x)W2(x)aW(x)dx=(1)1(x)(((i33)(188+12249)1/3+47i32+9i832+92492+1412)x2i3(188+12383)1/3+9i83+47i33383+(188+12383)2/3+2(188+12383)1/3473(188+12383)2/3x(2i3+2)(188+12383)1/3+(188+12383)2/3+(9i33)8347i3473(188+12383)2/3((i3+3)(188+12249)1/3+47i32+9i832924921412)3(188+12249)2/3)(x3)(832x)dx=x(((i33)(188+12249)1/3+47i32+9i832+92492+1412)x2i3(188+12383)1/3+9i83+47i33383+(188+12383)2/3+2(188+12383)1/3473(188+12383)2/3x(2i3+2)(188+12383)1/3+(188+12383)2/3+(9i33)8347i3473(188+12383)2/3((i3+3)(188+12249)1/3+47i32+9i832924921412))3(188+12249)2/3x2832dx=(2(((i33)(188+12249)1/3+47i32+9i832+92492+1412)x2i3(188+12383)1/3+9i83+47i33383+(188+12383)2/3+2(188+12383)1/3473(188+12383)2/3x(2i3+2)(188+12383)1/3+(188+12383)2/3+(9i33)8347i3473(188+12383)2/3((i3+3)(188+12249)1/3+47i32+9i832924921412))83249x(188+12249)2/3)dx=(383332i383332+i(188+12383)2/396i(188+12383)2/33832656+383(188+12383)2/326563(188+12383)2/396i(188+12383)1/36i12312)e((2i3+2)(188+12383)1/3+(188+12383)2/3+(9i33)8347i347)ln(x)3(188+12383)2/3+(383332+i383332i(188+12383)2/396+i(188+12383)2/33832656+383(188+12383)2/326563(188+12383)2/396+i(188+12383)1/36+i12312)e(2i3(188+12383)1/3+9i83+47i33383+(188+12383)2/3+2(188+12383)1/347)ln(x)3(188+12383)2/3(188+12383)1/3
U3=(1)33F(x)W3(x)aW(x)dx=(1)0(x)(3((188+12383)2/3+8)x(188+12383)2/3+2(188+12383)1/386(188+12383)1/312(188+12383)1/3)(x3)(832x)dx=x3((188+12383)2/3+8)x(188+12383)2/3+2(188+12383)1/386(188+12383)1/312(188+12383)1/3x2832dx=(x(188+12249)2/34(188+12249)1/386(188+12249)1/3((188+12249)2/3+8)249498(188+12249)1/3)dx=x1+(188+12249)2/34(188+12249)1/386(188+12249)1/3249((188+12249)2/3+8)(249(188+12249)2/313(188+12249)2/3+4(188+12249)1/324968(188+12249)1/3+32)95616(188+12249)1/3=x1+(188+12249)2/34(188+12249)1/386(188+12249)1/3249((188+12249)2/3+8)(249(188+12249)2/313(188+12249)2/3+4(188+12249)1/324968(188+12249)1/3+32)95616(188+12249)1/3

Now that all the Ui functions have been determined, the particular solution is found from

yp=U1y1+U2y2+U3y3

Hence

yp=((3i83332+383332i(188+12383)2/3396(188+12383)2/396+3i(188+12383)2/3832656+(188+12383)2/33832656+(188+12383)1/36i312+112)e((2i3+2)(188+12383)1/3+(188+12383)2/3+(9i33)8347i347)ln(x)3(188+12383)2/3+(3i83332+383332+i(188+12383)2/3396(188+12383)2/3963i(188+12383)2/3832656+(188+12383)2/33832656+(188+12383)1/36+i312+112)e(2i3(188+12383)1/3+9i83+47i33383+(188+12383)2/3+2(188+12383)1/347)ln(x)3(188+12383)2/3(188+12383)1/3)(x(188+12249)1/31223(188+12249)1/3+23cos(3((188+12249)1/3643(188+12249)1/3)ln(x)2))+((383332i383332+i(188+12383)2/396i(188+12383)2/33832656+383(188+12383)2/326563(188+12383)2/396i(188+12383)1/36i12312)e((2i3+2)(188+12383)1/3+(188+12383)2/3+(9i33)8347i347)ln(x)3(188+12383)2/3+(383332+i383332i(188+12383)2/396+i(188+12383)2/33832656+383(188+12383)2/326563(188+12383)2/396+i(188+12383)1/36+i12312)e(2i3(188+12383)1/3+9i83+47i33383+(188+12383)2/3+2(188+12383)1/347)ln(x)3(188+12383)2/3(188+12383)1/3)(x(188+12249)1/31223(188+12249)1/3+23sin(3((188+12249)1/3643(188+12249)1/3)ln(x)2))+(x1+(188+12249)2/34(188+12249)1/386(188+12249)1/3249((188+12249)2/3+8)(249(188+12249)2/313(188+12249)2/3+4(188+12249)1/324968(188+12249)1/3+32)95616(188+12249)1/3)(x(188+12249)1/36+43(188+12249)1/3+23)

Therefore the particular solution is

yp=x2

Therefore the general solution is

y=yh+yp=(x(188+12249)1/31223(188+12249)1/3+23(c1cos(3((188+12249)1/3643(188+12249)1/3)ln(x)2)c2sin(3((188+12249)1/3643(188+12249)1/3)ln(x)2))+c3x(188+12249)1/36+43(188+12249)1/3+23)+(x2)

Maple. Time used: 0.006 (sec). Leaf size: 223
ode:=x^4*diff(diff(diff(y(x),x),x),x)+x^3*diff(diff(y(x),x),x)+x^2*diff(y(x),x)+x*y(x) = x; 
dsolve(ode,y(x), singsol=all);
 
y=c2x(3249+47)(188+12249)2/3192+(188+12249)1/312+23cos((188+12383)1/33(3(188+12383)1/338347(188+12383)1/3+16)ln(x)192)+c3x(3249+47)(188+12249)2/3192+(188+12249)1/312+23sin((188+12383)1/33(3(188+12383)1/338347(188+12383)1/3+16)ln(x)192)+c1x(324947)(188+12249)2/396(188+12249)1/36+23+1

Maple trace

Methods for third order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying high order exact linear fully integrable 
trying differential order: 3; linear nonhomogeneous with symmetry [0,1] 
trying high order linear exact nonhomogeneous 
trying differential order: 3; missing the dependent variable 
checking if the LODE is of Euler type 
<- LODE of Euler type successful 
Euler equation successful
 

Maple step by step

Let’s solvex4(ddxd2dx2y(x))+x3(ddxddxy(x))+x2(ddxy(x))+xy(x)=xHighest derivative means the order of the ODE is3ddxd2dx2y(x)
Mathematica. Time used: 0.004 (sec). Leaf size: 82
ode=x^4*D[y[x],{x,3}]+x^3*D[y[x],{x,2}]+x^2*D[y[x],x]+x*y[x]== x; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 
y(x)c1xRoot[#132#12+2#1+1&,1]+c3xRoot[#132#12+2#1+1&,3]+c2xRoot[#132#12+2#1+1&,2]+1
Sympy. Time used: 1.379 (sec). Leaf size: 233
from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**4*Derivative(y(x), (x, 3)) + x**3*Derivative(y(x), (x, 2)) + x**2*Derivative(y(x), x) + x*y(x) - x,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 
y(x)=C1x23223347+32493+22347+324936+C2x23347+32493+22347+3249312+23sin(233(447+32493+2347+32493)log(x)12)+C3x23347+32493+22347+3249312+23cos(233(447+32493+2347+32493)log(x)12)+1