2.29 problem 28

Internal problem ID [7813]
Book : Own collection of miscellaneous problems
Section : section 2.0
Problem number : 28
Date solved : Monday, October 21, 2024 at 04:21:37 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

Solve

\begin{align*} y^{\prime \prime }-x y-x^{3}+2&=0 \end{align*}

2.29.1 Solved as second order Airy ode

Time used: 0.052 (sec)

This is Airy ODE. It has the general form

\[ a y^{\prime \prime } + b y^{\prime } + c x y = F(x) \]

Where in this case

\begin{align*} a &= 1\\ b &= 0\\ c &= -1\\ F &= x^{3}-2 \end{align*}

Therefore the solution to the homogeneous Airy ODE becomes

\[ y = c_1 \operatorname {AiryAi}\left (-x \left (-1\right )^{{1}/{3}}\right )+c_2 \operatorname {AiryBi}\left (-x \left (-1\right )^{{1}/{3}}\right ) \]

Since this is inhomogeneous Airy ODE, then we need to find the particular solution. The particular solution is now found using the method of undetermined coefficients. Looking at the RHS of the ode, which is

\[ x^{3}+1 \]

Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is

\[ [\{1, x, x^{2}, x^{3}\}] \]

While the set of the basis functions for the homogeneous solution found earlier is

\[ \left \{\operatorname {AiryAi}\left (-x \left (-1\right )^{{1}/{3}}\right ), \operatorname {AiryBi}\left (-x \left (-1\right )^{{1}/{3}}\right )\right \} \]

Since there is no duplication between the basis function in the UC_set and the basis functions of the homogeneous solution, the trial solution is a linear combination of all the basis in the UC_set.

\[ y_p = A_{4} x^{3}+A_{3} x^{2}+A_{2} x +A_{1} \]

The unknowns \(\{A_{1}, A_{2}, A_{3}, A_{4}\}\) 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

\[ 6 x A_{4}+2 A_{3}-x \left (A_{4} x^{3}+A_{3} x^{2}+A_{2} x +A_{1}\right )-x^{3}+2 = 0 \]

Solving for the unknowns by comparing coefficients results in

\[ [A_{1} = 0, A_{2} = 0, A_{3} = -1, A_{4} = 0] \]

Substituting the above back in the above trial solution \(y_p\), gives the particular solution

\[ y_p = -x^{2} \]

Therefore the general solution is

\begin{align*} y &= y_h + y_p \\ &= \left (c_1 \operatorname {AiryAi}\left (-x \left (-1\right )^{{1}/{3}}\right )+c_2 \operatorname {AiryBi}\left (-x \left (-1\right )^{{1}/{3}}\right )\right ) + \left (-x^{2}\right ) \\ &= c_1 \operatorname {AiryAi}\left (-x \left (-1\right )^{{1}/{3}}\right )+c_2 \operatorname {AiryBi}\left (-x \left (-1\right )^{{1}/{3}}\right )-x^{2} \\ \end{align*}

Will add steps showing solving for IC soon.

2.29.2 Solved as second order ode adjoint method

Time used: 0.946 (sec)

In normal form the ode

\begin{align*} y^{\prime \prime }-x y-x^{3}+2 = 0 \tag {1} \end{align*}

Becomes

\begin{align*} y^{\prime \prime }+p \left (x \right ) y^{\prime }+q \left (x \right ) y&=r \left (x \right ) \tag {2} \end{align*}

Where

\begin{align*} p \left (x \right )&=0\\ q \left (x \right )&=-x\\ r \left (x \right )&=x^{3}-2 \end{align*}

The Lagrange adjoint ode is given by

\begin{align*} \xi ^{''}-(\xi \, p)'+\xi q &= 0\\ \xi ^{''}-\left (0\right )' + \left (-x \xi \left (x \right )\right ) &= 0\\ \xi ^{\prime \prime }\left (x \right )-x \xi \left (x \right )&= 0 \end{align*}

Which is solved for \(\xi (x)\). This is Airy ODE. It has the general form

\[ a \xi ^{\prime \prime } + b \xi ^{\prime } + c x \xi = F(x) \]

Where in this case

\begin{align*} a &= 1\\ b &= 0\\ c &= -1\\ F &= 0 \end{align*}

Therefore the solution to the homogeneous Airy ODE becomes

\[ \xi = c_5 \operatorname {AiryAi}\left (-x \left (-1\right )^{{1}/{3}}\right )+c_6 \operatorname {AiryBi}\left (-x \left (-1\right )^{{1}/{3}}\right ) \]

Will add steps showing solving for IC soon.

The original ode (2) now reduces to first order ode

\begin{align*} \xi \left (x \right ) y^{\prime }-y \xi ^{\prime }\left (x \right )+\xi \left (x \right ) p \left (x \right ) y&=\int \xi \left (x \right ) r \left (x \right )d x\\ y^{\prime }+y \left (p \left (x \right )-\frac {\xi ^{\prime }\left (x \right )}{\xi \left (x \right )}\right )&=\frac {\int \xi \left (x \right ) r \left (x \right )d x}{\xi \left (x \right )}\\ y^{\prime }-\frac {y \left (-c_5 \left (-1\right )^{{1}/{3}} \operatorname {AiryAi}\left (1, -x \left (-1\right )^{{1}/{3}}\right )-c_6 \left (-1\right )^{{1}/{3}} \operatorname {AiryBi}\left (1, -x \left (-1\right )^{{1}/{3}}\right )\right )}{c_5 \operatorname {AiryAi}\left (-x \left (-1\right )^{{1}/{3}}\right )+c_6 \operatorname {AiryBi}\left (-x \left (-1\right )^{{1}/{3}}\right )}&=\frac {-2 c_5 x \operatorname {AiryAi}\left (-x \left (-1\right )^{{1}/{3}}\right )-2 c_6 x \operatorname {AiryBi}\left (-x \left (-1\right )^{{1}/{3}}\right )-\frac {c_5 \operatorname {AiryAi}\left (1, -x \left (-1\right )^{{1}/{3}}\right ) x^{2} \sqrt {-3}}{2}-\frac {c_5 \operatorname {AiryAi}\left (1, -x \left (-1\right )^{{1}/{3}}\right ) x^{2}}{2}-\frac {c_6 \operatorname {AiryBi}\left (1, -x \left (-1\right )^{{1}/{3}}\right ) x^{2} \sqrt {-3}}{2}-\frac {c_6 \operatorname {AiryBi}\left (1, -x \left (-1\right )^{{1}/{3}}\right ) x^{2}}{2}}{c_5 \operatorname {AiryAi}\left (-x \left (-1\right )^{{1}/{3}}\right )+c_6 \operatorname {AiryBi}\left (-x \left (-1\right )^{{1}/{3}}\right )} \end{align*}

Which is now a first order ode. This is now solved for \(y\). In canonical form a linear first order is

\begin{align*} y^{\prime } + q(x)y &= p(x) \end{align*}

Comparing the above to the given ode shows that

\begin{align*} q(x) &=\frac {2 c_5 \left (-1\right )^{{1}/{3}} \operatorname {AiryAi}\left (1, -x \left (-1\right )^{{1}/{3}}\right )+2 c_6 \left (-1\right )^{{1}/{3}} \operatorname {AiryBi}\left (1, -x \left (-1\right )^{{1}/{3}}\right )}{2 c_5 \operatorname {AiryAi}\left (-x \left (-1\right )^{{1}/{3}}\right )+2 c_6 \operatorname {AiryBi}\left (-x \left (-1\right )^{{1}/{3}}\right )}\\ p(x) &=-\frac {c_5 \operatorname {AiryAi}\left (1, -x \left (-1\right )^{{1}/{3}}\right ) x^{2} \sqrt {-3}+c_6 \operatorname {AiryBi}\left (1, -x \left (-1\right )^{{1}/{3}}\right ) x^{2} \sqrt {-3}+c_5 \operatorname {AiryAi}\left (1, -x \left (-1\right )^{{1}/{3}}\right ) x^{2}+c_6 \operatorname {AiryBi}\left (1, -x \left (-1\right )^{{1}/{3}}\right ) x^{2}+4 c_5 x \operatorname {AiryAi}\left (-x \left (-1\right )^{{1}/{3}}\right )+4 c_6 x \operatorname {AiryBi}\left (-x \left (-1\right )^{{1}/{3}}\right )}{2 \left (c_5 \operatorname {AiryAi}\left (-x \left (-1\right )^{{1}/{3}}\right )+c_6 \operatorname {AiryBi}\left (-x \left (-1\right )^{{1}/{3}}\right )\right )} \end{align*}

The integrating factor \(\mu \) is

\begin{align*} \mu &= e^{\int {q\,dx}}\\ &= {\mathrm e}^{\int \frac {2 c_5 \left (-1\right )^{{1}/{3}} \operatorname {AiryAi}\left (1, -x \left (-1\right )^{{1}/{3}}\right )+2 c_6 \left (-1\right )^{{1}/{3}} \operatorname {AiryBi}\left (1, -x \left (-1\right )^{{1}/{3}}\right )}{2 c_5 \operatorname {AiryAi}\left (-x \left (-1\right )^{{1}/{3}}\right )+2 c_6 \operatorname {AiryBi}\left (-x \left (-1\right )^{{1}/{3}}\right )}d x}\\ &= \frac {1}{c_5 \operatorname {AiryAi}\left (-\frac {\left (1+i \sqrt {3}\right ) x}{2}\right )+c_6 \operatorname {AiryBi}\left (-\frac {\left (1+i \sqrt {3}\right ) x}{2}\right )} \end{align*}

The ode becomes

\begin{align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}}\left ( \mu y\right ) &= \mu p \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}}\left ( \mu y\right ) &= \left (\mu \right ) \left (-\frac {c_5 \operatorname {AiryAi}\left (1, -x \left (-1\right )^{{1}/{3}}\right ) x^{2} \sqrt {-3}+c_6 \operatorname {AiryBi}\left (1, -x \left (-1\right )^{{1}/{3}}\right ) x^{2} \sqrt {-3}+c_5 \operatorname {AiryAi}\left (1, -x \left (-1\right )^{{1}/{3}}\right ) x^{2}+c_6 \operatorname {AiryBi}\left (1, -x \left (-1\right )^{{1}/{3}}\right ) x^{2}+4 c_5 x \operatorname {AiryAi}\left (-x \left (-1\right )^{{1}/{3}}\right )+4 c_6 x \operatorname {AiryBi}\left (-x \left (-1\right )^{{1}/{3}}\right )}{2 \left (c_5 \operatorname {AiryAi}\left (-x \left (-1\right )^{{1}/{3}}\right )+c_6 \operatorname {AiryBi}\left (-x \left (-1\right )^{{1}/{3}}\right )\right )}\right ) \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}} \left (\frac {y}{c_5 \operatorname {AiryAi}\left (-\frac {\left (1+i \sqrt {3}\right ) x}{2}\right )+c_6 \operatorname {AiryBi}\left (-\frac {\left (1+i \sqrt {3}\right ) x}{2}\right )}\right ) &= \left (\frac {1}{c_5 \operatorname {AiryAi}\left (-\frac {\left (1+i \sqrt {3}\right ) x}{2}\right )+c_6 \operatorname {AiryBi}\left (-\frac {\left (1+i \sqrt {3}\right ) x}{2}\right )}\right ) \left (-\frac {c_5 \operatorname {AiryAi}\left (1, -x \left (-1\right )^{{1}/{3}}\right ) x^{2} \sqrt {-3}+c_6 \operatorname {AiryBi}\left (1, -x \left (-1\right )^{{1}/{3}}\right ) x^{2} \sqrt {-3}+c_5 \operatorname {AiryAi}\left (1, -x \left (-1\right )^{{1}/{3}}\right ) x^{2}+c_6 \operatorname {AiryBi}\left (1, -x \left (-1\right )^{{1}/{3}}\right ) x^{2}+4 c_5 x \operatorname {AiryAi}\left (-x \left (-1\right )^{{1}/{3}}\right )+4 c_6 x \operatorname {AiryBi}\left (-x \left (-1\right )^{{1}/{3}}\right )}{2 \left (c_5 \operatorname {AiryAi}\left (-x \left (-1\right )^{{1}/{3}}\right )+c_6 \operatorname {AiryBi}\left (-x \left (-1\right )^{{1}/{3}}\right )\right )}\right ) \\ \mathrm {d} \left (\frac {y}{c_5 \operatorname {AiryAi}\left (-\frac {\left (1+i \sqrt {3}\right ) x}{2}\right )+c_6 \operatorname {AiryBi}\left (-\frac {\left (1+i \sqrt {3}\right ) x}{2}\right )}\right ) &= \left (-\frac {c_5 \operatorname {AiryAi}\left (1, -x \left (-1\right )^{{1}/{3}}\right ) x^{2} \sqrt {-3}+c_6 \operatorname {AiryBi}\left (1, -x \left (-1\right )^{{1}/{3}}\right ) x^{2} \sqrt {-3}+c_5 \operatorname {AiryAi}\left (1, -x \left (-1\right )^{{1}/{3}}\right ) x^{2}+c_6 \operatorname {AiryBi}\left (1, -x \left (-1\right )^{{1}/{3}}\right ) x^{2}+4 c_5 x \operatorname {AiryAi}\left (-x \left (-1\right )^{{1}/{3}}\right )+4 c_6 x \operatorname {AiryBi}\left (-x \left (-1\right )^{{1}/{3}}\right )}{2 \left (c_5 \operatorname {AiryAi}\left (-x \left (-1\right )^{{1}/{3}}\right )+c_6 \operatorname {AiryBi}\left (-x \left (-1\right )^{{1}/{3}}\right )\right ) \left (c_5 \operatorname {AiryAi}\left (-\frac {\left (1+i \sqrt {3}\right ) x}{2}\right )+c_6 \operatorname {AiryBi}\left (-\frac {\left (1+i \sqrt {3}\right ) x}{2}\right )\right )}\right )\, \mathrm {d} x \\ \end{align*}

Integrating gives

\begin{align*} \frac {y}{c_5 \operatorname {AiryAi}\left (-\frac {\left (1+i \sqrt {3}\right ) x}{2}\right )+c_6 \operatorname {AiryBi}\left (-\frac {\left (1+i \sqrt {3}\right ) x}{2}\right )}&= \int {-\frac {c_5 \operatorname {AiryAi}\left (1, -x \left (-1\right )^{{1}/{3}}\right ) x^{2} \sqrt {-3}+c_6 \operatorname {AiryBi}\left (1, -x \left (-1\right )^{{1}/{3}}\right ) x^{2} \sqrt {-3}+c_5 \operatorname {AiryAi}\left (1, -x \left (-1\right )^{{1}/{3}}\right ) x^{2}+c_6 \operatorname {AiryBi}\left (1, -x \left (-1\right )^{{1}/{3}}\right ) x^{2}+4 c_5 x \operatorname {AiryAi}\left (-x \left (-1\right )^{{1}/{3}}\right )+4 c_6 x \operatorname {AiryBi}\left (-x \left (-1\right )^{{1}/{3}}\right )}{2 \left (c_5 \operatorname {AiryAi}\left (-x \left (-1\right )^{{1}/{3}}\right )+c_6 \operatorname {AiryBi}\left (-x \left (-1\right )^{{1}/{3}}\right )\right ) \left (c_5 \operatorname {AiryAi}\left (-\frac {\left (1+i \sqrt {3}\right ) x}{2}\right )+c_6 \operatorname {AiryBi}\left (-\frac {\left (1+i \sqrt {3}\right ) x}{2}\right )\right )} \,dx} \\ &=\int -\frac {c_5 \operatorname {AiryAi}\left (1, -x \left (-1\right )^{{1}/{3}}\right ) x^{2} \sqrt {-3}+c_6 \operatorname {AiryBi}\left (1, -x \left (-1\right )^{{1}/{3}}\right ) x^{2} \sqrt {-3}+c_5 \operatorname {AiryAi}\left (1, -x \left (-1\right )^{{1}/{3}}\right ) x^{2}+c_6 \operatorname {AiryBi}\left (1, -x \left (-1\right )^{{1}/{3}}\right ) x^{2}+4 c_5 x \operatorname {AiryAi}\left (-x \left (-1\right )^{{1}/{3}}\right )+4 c_6 x \operatorname {AiryBi}\left (-x \left (-1\right )^{{1}/{3}}\right )}{2 \left (c_5 \operatorname {AiryAi}\left (-x \left (-1\right )^{{1}/{3}}\right )+c_6 \operatorname {AiryBi}\left (-x \left (-1\right )^{{1}/{3}}\right )\right ) \left (c_5 \operatorname {AiryAi}\left (-\frac {\left (1+i \sqrt {3}\right ) x}{2}\right )+c_6 \operatorname {AiryBi}\left (-\frac {\left (1+i \sqrt {3}\right ) x}{2}\right )\right )}d x + c_7 \end{align*}

Dividing throughout by the integrating factor \(\frac {1}{c_5 \operatorname {AiryAi}\left (-\frac {\left (1+i \sqrt {3}\right ) x}{2}\right )+c_6 \operatorname {AiryBi}\left (-\frac {\left (1+i \sqrt {3}\right ) x}{2}\right )}\) gives the final solution

\[ y = \left (c_5 \operatorname {AiryAi}\left (-\frac {\left (1+i \sqrt {3}\right ) x}{2}\right )+c_6 \operatorname {AiryBi}\left (-\frac {\left (1+i \sqrt {3}\right ) x}{2}\right )\right ) \left (\int -\frac {c_5 \operatorname {AiryAi}\left (1, -x \left (-1\right )^{{1}/{3}}\right ) x^{2} \sqrt {-3}+c_6 \operatorname {AiryBi}\left (1, -x \left (-1\right )^{{1}/{3}}\right ) x^{2} \sqrt {-3}+c_5 \operatorname {AiryAi}\left (1, -x \left (-1\right )^{{1}/{3}}\right ) x^{2}+c_6 \operatorname {AiryBi}\left (1, -x \left (-1\right )^{{1}/{3}}\right ) x^{2}+4 c_5 x \operatorname {AiryAi}\left (-x \left (-1\right )^{{1}/{3}}\right )+4 c_6 x \operatorname {AiryBi}\left (-x \left (-1\right )^{{1}/{3}}\right )}{2 \left (c_5 \operatorname {AiryAi}\left (-x \left (-1\right )^{{1}/{3}}\right )+c_6 \operatorname {AiryBi}\left (-x \left (-1\right )^{{1}/{3}}\right )\right ) \left (c_5 \operatorname {AiryAi}\left (-\frac {\left (1+i \sqrt {3}\right ) x}{2}\right )+c_6 \operatorname {AiryBi}\left (-\frac {\left (1+i \sqrt {3}\right ) x}{2}\right )\right )}d x +c_7 \right ) \]

Hence, the solution found using Lagrange adjoint equation method is

\begin{align*} y &= \left (c_5 \operatorname {AiryAi}\left (-\frac {\left (1+i \sqrt {3}\right ) x}{2}\right )+c_6 \operatorname {AiryBi}\left (-\frac {\left (1+i \sqrt {3}\right ) x}{2}\right )\right ) \left (\int -\frac {c_5 \operatorname {AiryAi}\left (1, -x \left (-1\right )^{{1}/{3}}\right ) x^{2} \sqrt {-3}+c_6 \operatorname {AiryBi}\left (1, -x \left (-1\right )^{{1}/{3}}\right ) x^{2} \sqrt {-3}+c_5 \operatorname {AiryAi}\left (1, -x \left (-1\right )^{{1}/{3}}\right ) x^{2}+c_6 \operatorname {AiryBi}\left (1, -x \left (-1\right )^{{1}/{3}}\right ) x^{2}+4 c_5 x \operatorname {AiryAi}\left (-x \left (-1\right )^{{1}/{3}}\right )+4 c_6 x \operatorname {AiryBi}\left (-x \left (-1\right )^{{1}/{3}}\right )}{2 \left (c_5 \operatorname {AiryAi}\left (-x \left (-1\right )^{{1}/{3}}\right )+c_6 \operatorname {AiryBi}\left (-x \left (-1\right )^{{1}/{3}}\right )\right ) \left (c_5 \operatorname {AiryAi}\left (-\frac {\left (1+i \sqrt {3}\right ) x}{2}\right )+c_6 \operatorname {AiryBi}\left (-\frac {\left (1+i \sqrt {3}\right ) x}{2}\right )\right )}d x +c_7 \right ) \\ \end{align*}

Will add steps showing solving for IC soon.

2.29.3 Maple step by step solution

2.29.4 Maple trace

Methods for second order ODEs:
 

2.29.5 Maple dsolve solution

Solving time : 0.002 (sec)
Leaf size : 18

dsolve(diff(diff(y(x),x),x)-x*y(x)-x^3+2 = 0, 
       y(x),singsol=all)
 

\[ y = \operatorname {AiryAi}\left (x \right ) c_2 +\operatorname {AiryBi}\left (x \right ) c_1 -x^{2} \]

2.29.6 Mathematica DSolve solution

Solving time : 0.456 (sec)
Leaf size : 290

DSolve[{D[y[x],{x,2}]-x*y[x]-x^3+2==0,{}}, 
       y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {6 \sqrt [3]{3} \pi x \operatorname {Gamma}\left (\frac {1}{3}\right ) \operatorname {Gamma}\left (\frac {5}{3}\right ) \operatorname {Gamma}\left (\frac {7}{3}\right ) \operatorname {Gamma}\left (\frac {8}{3}\right ) \left (\sqrt {3} \operatorname {AiryAi}(x)-\operatorname {AiryBi}(x)\right ) \, _1F_2\left (\frac {1}{3};\frac {2}{3},\frac {4}{3};\frac {x^3}{9}\right )+2 \sqrt [6]{3} \pi x^2 \operatorname {Gamma}\left (\frac {2}{3}\right )^2 \operatorname {Gamma}\left (\frac {7}{3}\right ) \operatorname {Gamma}\left (\frac {8}{3}\right ) \left (3 \operatorname {AiryAi}(x)+\sqrt {3} \operatorname {AiryBi}(x)\right ) \, _1F_2\left (\frac {2}{3};\frac {4}{3},\frac {5}{3};\frac {x^3}{9}\right )+\operatorname {Gamma}\left (\frac {5}{3}\right ) \left (3 \sqrt [3]{3} \pi x^4 \operatorname {Gamma}\left (\frac {4}{3}\right )^2 \operatorname {Gamma}\left (\frac {8}{3}\right ) \left (\operatorname {AiryBi}(x)-\sqrt {3} \operatorname {AiryAi}(x)\right ) \, _1F_2\left (\frac {4}{3};\frac {2}{3},\frac {7}{3};\frac {x^3}{9}\right )+\operatorname {Gamma}\left (\frac {2}{3}\right ) \operatorname {Gamma}\left (\frac {7}{3}\right ) \left (-\sqrt [6]{3} \pi x^5 \operatorname {Gamma}\left (\frac {5}{3}\right ) \left (3 \operatorname {AiryAi}(x)+\sqrt {3} \operatorname {AiryBi}(x)\right ) \, _1F_2\left (\frac {5}{3};\frac {4}{3},\frac {8}{3};\frac {x^3}{9}\right )+27 \operatorname {Gamma}\left (\frac {4}{3}\right ) \operatorname {Gamma}\left (\frac {8}{3}\right ) (c_1 \operatorname {AiryAi}(x)+c_2 \operatorname {AiryBi}(x))\right )\right )}{27 \operatorname {Gamma}\left (\frac {2}{3}\right ) \operatorname {Gamma}\left (\frac {4}{3}\right ) \operatorname {Gamma}\left (\frac {5}{3}\right ) \operatorname {Gamma}\left (\frac {7}{3}\right ) \operatorname {Gamma}\left (\frac {8}{3}\right )} \]