2.42 problem 42
Internal
problem
ID
[8131]
Book
:
Second
order
enumerated
odes
Section
:
section
2
Problem
number
:
42
Date
solved
:
Monday, October 21, 2024 at 04:54:10 PM
CAS
classification
:
[[_high_order, _linear, _nonhomogeneous]]
Solve
\begin{align*} y^{\prime \prime \prime \prime }-y^{\prime \prime \prime }-3 y^{\prime \prime }+5 y^{\prime }-2 y&=x \,{\mathrm e}^{x}+3 \,{\mathrm e}^{-2 x} \end{align*}
2.42.1 Solved as higher order constant coeff ode
Time used: 0.167 (sec)
The characteristic equation is
\[ \lambda ^{4}-\lambda ^{3}-3 \lambda ^{2}+5 \lambda -2 = 0 \]
The roots of the above equation are
\begin{align*} \lambda _1 &= -2\\ \lambda _2 &= 1\\ \lambda _3 &= 1\\ \lambda _4 &= 1 \end{align*}
Therefore the homogeneous solution is
\[ y_h(x)={\mathrm e}^{-2 x} c_1 +{\mathrm e}^{x} c_2 +x \,{\mathrm e}^{x} c_3 +x^{2} {\mathrm e}^{x} c_4 \]
The fundamental set of solutions for the
homogeneous solution are the following
\begin{align*} y_1 &= {\mathrm e}^{-2 x}\\ y_2 &= {\mathrm e}^{x}\\ y_3 &= x \,{\mathrm e}^{x}\\ y_4 &= x^{2} {\mathrm e}^{x} \end{align*}
This is higher order nonhomogeneous ODE. Let the solution be
\[ y = y_h + y_p \]
Where \(y_h\) is the solution to
the homogeneous ODE And \(y_p\) is a particular solution to the nonhomogeneous ODE. \(y_h\) is the
solution to
\[ y^{\prime \prime \prime \prime }-y^{\prime \prime \prime }-3 y^{\prime \prime }+5 y^{\prime }-2 y = 0 \]
Now the particular solution to the given ODE is found
\[
y^{\prime \prime \prime \prime }-y^{\prime \prime \prime }-3 y^{\prime \prime }+5 y^{\prime }-2 y = \left (x \,{\mathrm e}^{3 x}+3\right ) {\mathrm e}^{-2 x}
\]
The particular solution
is now found using the method of undetermined coefficients.
Looking at the RHS of the ode, which is
\[ \left (x \,{\mathrm e}^{3 x}+3\right ) {\mathrm e}^{-2 x} \]
Shows that the corresponding undetermined set of
the basis functions (UC_set) for the trial solution is
\[ [\{{\mathrm e}^{-2 x}\}, \{x \,{\mathrm e}^{x}, {\mathrm e}^{x}\}] \]
While the set of the basis functions for
the homogeneous solution found earlier is
\[ \{x \,{\mathrm e}^{x}, x^{2} {\mathrm e}^{x}, {\mathrm e}^{x}, {\mathrm e}^{-2 x}\} \]
Since \({\mathrm e}^{-2 x}\) is duplicated in the UC_set, then this basis
is multiplied by extra \(x\). The UC_set becomes
\[ [\{x \,{\mathrm e}^{-2 x}\}, \{x \,{\mathrm e}^{x}, {\mathrm e}^{x}\}] \]
Since \({\mathrm e}^{x}\) is duplicated in the UC_set, then this
basis is multiplied by extra \(x\). The UC_set becomes
\[ [\{x \,{\mathrm e}^{-2 x}\}, \{x \,{\mathrm e}^{x}, x^{2} {\mathrm e}^{x}\}] \]
Since \(x \,{\mathrm e}^{x}\) is duplicated in the UC_set, then
this basis is multiplied by extra \(x\). The UC_set becomes
\[ [\{x \,{\mathrm e}^{-2 x}\}, \{x^{2} {\mathrm e}^{x}, x^{3} {\mathrm e}^{x}\}] \]
Since \(x^{2} {\mathrm e}^{x}\) is duplicated in
the UC_set, then this basis is multiplied by extra \(x\). The UC_set becomes
\[ [\{x \,{\mathrm e}^{-2 x}\}, \{x^{3} {\mathrm e}^{x}, x^{4} {\mathrm e}^{x}\}] \]
Since
there was duplication between the basis functions in the UC_set and the basis
functions of the homogeneous solution, the trial solution is a linear combination of all
the basis function in the above updated UC_set.
\[
y_p = A_{1} x \,{\mathrm e}^{-2 x}+A_{2} x^{3} {\mathrm e}^{x}+A_{3} x^{4} {\mathrm e}^{x}
\]
The unknowns \(\{A_{1}, A_{2}, A_{3}\}\) are found by
substituting the above trial solution \(y_p\) into the ODE and comparing coefficients.
Substituting the trial solution into the ODE and simplifying gives
\[
72 A_{3} x \,{\mathrm e}^{x}+18 A_{2} {\mathrm e}^{x}+24 A_{3} {\mathrm e}^{x}-27 A_{1} {\mathrm e}^{-2 x} = x \,{\mathrm e}^{x}+3 \,{\mathrm e}^{-2 x}
\]
Solving for the
unknowns by comparing coefficients results in
\[ \left [A_{1} = -{\frac {1}{9}}, A_{2} = -{\frac {1}{54}}, A_{3} = {\frac {1}{72}}\right ] \]
Substituting the above back in the
above trial solution \(y_p\), gives the particular solution
\[
y_p = -\frac {x \,{\mathrm e}^{-2 x}}{9}-\frac {x^{3} {\mathrm e}^{x}}{54}+\frac {x^{4} {\mathrm e}^{x}}{72}
\]
Therefore the general solution is
\begin{align*}
y &= y_h + y_p \\
&= \left ({\mathrm e}^{-2 x} c_1 +{\mathrm e}^{x} c_2 +x \,{\mathrm e}^{x} c_3 +x^{2} {\mathrm e}^{x} c_4\right ) + \left (-\frac {x \,{\mathrm e}^{-2 x}}{9}-\frac {x^{3} {\mathrm e}^{x}}{54}+\frac {x^{4} {\mathrm e}^{x}}{72}\right ) \\
\end{align*}
2.42.2 Maple step by step solution
2.42.3 Maple trace
Methods for high order ODEs:
2.42.4 Maple dsolve solution
Solving time : 0.006
(sec)
Leaf size : 52
dsolve(diff(diff(diff(diff(y(x),x),x),x),x)-diff(diff(diff(y(x),x),x),x)-3*diff(diff(y(x),x),x)+5*diff(y(x),x)-2*y(x) = x*exp(x)+3*exp(-2*x),
y(x),singsol=all)
\[
y = \frac {\left (\left (x^{4}-\frac {4 x^{3}}{3}+\left (72 c_4 +\frac {4}{3}\right ) x^{2}+\left (72 c_3 -\frac {8}{9}\right ) x +72 c_1 +\frac {8}{27}\right ) {\mathrm e}^{3 x}-8 x +72 c_2 -8\right ) {\mathrm e}^{-2 x}}{72}
\]
2.42.5 Mathematica DSolve solution
Solving time : 0.408
(sec)
Leaf size : 64
DSolve[{D[y[x],{x,4}]-D[y[x],{x,3}]-3*D[y[x],{x,2}]+5*D[y[x],x]-2*y[x]==x*Exp[x]+3*Exp[-2*x],{}},
y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to e^x \left (\frac {x^4}{72}-\frac {x^3}{54}+\left (\frac {1}{54}+c_4\right ) x^2+\left (-\frac {1}{81}+c_3\right ) x+\frac {1}{243}+c_2\right )-\frac {1}{9} e^{-2 x} (x+1-9 c_1)
\]