problem 5

Internal problem ID [12747]
Internal ﬁle name [OUTPUT/11400_Friday_November_03_2023_06_32_25_AM_18586947/index.tex]

Book: Ordinary Diﬀerential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section: Chapter 4. N-th Order Linear Diﬀerential Equations. Exercises 4.1, page 186
Problem number: 5.
ODE order: 2.
ODE degree: 1.

\[ \boxed {\sqrt {1-x}\, y^{\prime \prime }-4 y=\sin \left (x \right )} \] With initial conditions \begin {align*} [y \left (-2\right ) = 3, y^{\prime }\left (-2\right ) = -1] \end {align*}

9.5.1 Existence and uniqueness analysis

This is a linear ODE. In canonical form it is written as \begin {align*} y^{\prime \prime } + p(x)y^{\prime } + q(x) y &= F \end {align*}

Where here \begin {align*} p(x) &=0\\ q(x) &=-\frac {4}{\sqrt {1-x}}\\ F &=\frac {\sin \left (x \right )}{\sqrt {1-x}} \end {align*}

Hence the ode is \begin {align*} y^{\prime \prime }-\frac {4 y}{\sqrt {1-x}} = \frac {\sin \left (x \right )}{\sqrt {1-x}} \end {align*}

9.5.2 Solving as second order bessel ode ode

Writing the ode as \begin {align*} x^{2} y^{\prime \prime }-4 x^{\frac {3}{2}} y = x^{\frac {3}{2}} \sin \left (x \right )\tag {1} \end {align*}

Where \(y_h\) is the solution to the homogeneous ODE and \(y_p\) is a particular solution to the non-homogeneous ODE. Bessel ode has the form \begin {align*} x^{2} y^{\prime \prime }+y^{\prime } x +\left (-n^{2}+x^{2}\right ) y = 0\tag {2} \end {align*}

The generalized form of Bessel ode is given by Bowman (1958) as the following \begin {align*} x^{2} y^{\prime \prime }+\left (1-2 \alpha \right ) x y^{\prime }+\left (\beta ^{2} \gamma ^{2} x^{2 \gamma }-n^{2} \gamma ^{2}+\alpha ^{2}\right ) y = 0\tag {3} \end {align*}

With the standard solution \begin {align*} y&=x^{\alpha } \left (c_{1} \operatorname {BesselJ}\left (n , \beta \,x^{\gamma }\right )+c_{2} \operatorname {BesselY}\left (n , \beta \,x^{\gamma }\right )\right )\tag {4} \end {align*}

Comparing (3) to (1) and solving for \(\alpha ,\beta ,n,\gamma \) gives \begin {align*} \alpha &= {\frac {1}{2}}\\ \beta &= \frac {8 i}{3}\\ n &= {\frac {2}{3}}\\ \gamma &= {\frac {3}{4}} \end {align*}

Substituting all the above into (4) gives the solution as \begin {align*} y = c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right ) \end {align*}

Therefore the homogeneous solution \(y_h\) is \[ y_h = c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right ) \] The particular solution \(y_p\) can be found using either the method of undetermined coeﬃcients, or the method of variation of parameters. The method of variation of parameters will be used as it is more general and can be used when the coeﬃcients of the ODE depend on \(x\) as well. Let \begin{equation} \tag{1} y_p(x) = u_1 y_1 + u_2 y_2 \end{equation} Where \(u_1,u_2\) to be determined, and \(y_1,y_2\) are the two basis solutions (the two linearly independent solutions of the homogeneous ODE) found earlier when solving the homogeneous ODE as \begin{align*} y_1 &= \sqrt {x}\, \operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right ) \\ y_2 &= \sqrt {x}\, \operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right ) \\ \end{align*} In the Variation of parameters \(u_1,u_2\) are found using \begin{align*} \tag{2} u_1 &= -\int \frac {y_2 f(x)}{a W(x)} \\ \tag{3} u_2 &= \int \frac {y_1 f(x)}{a W(x)} \\ \end{align*} Where \(W(x)\) is the Wronskian and \(a\) is the coeﬃcient in front of \(y''\) in the given ODE. The Wronskian is given by \(W= \begin {vmatrix} y_1 & y_{2} \\ y_{1}^{\prime } & y_{2}^{\prime } \end {vmatrix} \). Hence \[ W = \begin {vmatrix} \sqrt {x}\, \operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right ) & \sqrt {x}\, \operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right ) \\ \frac {d}{dx}\left (\sqrt {x}\, \operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )\right ) & \frac {d}{dx}\left (\sqrt {x}\, \operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )\right ) \end {vmatrix} \] Which gives \[ W = \begin {vmatrix} \sqrt {x}\, \operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right ) & \sqrt {x}\, \operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right ) \\ \frac {\operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )}{2 \sqrt {x}}+2 i x^{\frac {1}{4}} \left (\operatorname {BesselJ}\left (-\frac {1}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )+\frac {i \operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )}{4 x^{\frac {3}{4}}}\right ) & \frac {\operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )}{2 \sqrt {x}}+2 i x^{\frac {1}{4}} \left (\operatorname {BesselY}\left (-\frac {1}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )+\frac {i \operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )}{4 x^{\frac {3}{4}}}\right ) \end {vmatrix} \] Therefore \[ W = \left (\sqrt {x}\, \operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )\right )\left (\frac {\operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )}{2 \sqrt {x}}+2 i x^{\frac {1}{4}} \left (\operatorname {BesselY}\left (-\frac {1}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )+\frac {i \operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )}{4 x^{\frac {3}{4}}}\right )\right ) - \left (\sqrt {x}\, \operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )\right )\left (\frac {\operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )}{2 \sqrt {x}}+2 i x^{\frac {1}{4}} \left (\operatorname {BesselJ}\left (-\frac {1}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )+\frac {i \operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )}{4 x^{\frac {3}{4}}}\right )\right ) \] Which simpliﬁes to \[ W = 2 i x^{\frac {3}{4}} \left (\operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right ) \operatorname {BesselY}\left (-\frac {1}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )-\operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right ) \operatorname {BesselJ}\left (-\frac {1}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )\right ) \] Which simpliﬁes to \[ W = \frac {3}{2 \pi } \] Therefore Eq. (2) becomes \[ u_1 = -\int \frac {x^{2} \operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right ) \sin \left (x \right )}{\frac {3 x^{2}}{2 \pi }}\,dx \] Which simpliﬁes to \[ u_1 = - \int \frac {2 \operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right ) \sin \left (x \right ) \pi }{3}d x \] Hence \[ u_1 = -\left (\int _{0}^{x}\frac {2 \operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i \alpha ^{\frac {3}{4}}}{3}\right ) \sin \left (\alpha \right ) \pi }{3}d \alpha \right ) \] And Eq. (3) becomes \[ u_2 = \int \frac {x^{2} \operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right ) \sin \left (x \right )}{\frac {3 x^{2}}{2 \pi }}\,dx \] Which simpliﬁes to \[ u_2 = \int \frac {2 \operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right ) \sin \left (x \right ) \pi }{3}d x \] Hence \[ u_2 = \int _{0}^{x}\frac {2 \operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i \alpha ^{\frac {3}{4}}}{3}\right ) \sin \left (\alpha \right ) \pi }{3}d \alpha \] Which simpliﬁes to \begin{align*} u_1 &= -\frac {2 \pi \left (\int _{0}^{x}\operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i \alpha ^{\frac {3}{4}}}{3}\right ) \sin \left (\alpha \right )d \alpha \right )}{3} \\ u_2 &= \frac {2 \pi \left (\int _{0}^{x}\operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i \alpha ^{\frac {3}{4}}}{3}\right ) \sin \left (\alpha \right )d \alpha \right )}{3} \\ \end{align*} Therefore the particular solution, from equation (1) is \[ y_p(x) = -\frac {2 \pi \left (\int _{0}^{x}\operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i \alpha ^{\frac {3}{4}}}{3}\right ) \sin \left (\alpha \right )d \alpha \right ) \sqrt {x}\, \operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )}{3}+\frac {2 \pi \left (\int _{0}^{x}\operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i \alpha ^{\frac {3}{4}}}{3}\right ) \sin \left (\alpha \right )d \alpha \right ) \sqrt {x}\, \operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )}{3} \] Which simpliﬁes to \[ y_p(x) = \frac {2 \pi \sqrt {x}\, \left (-\left (\int _{0}^{x}\operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i \alpha ^{\frac {3}{4}}}{3}\right ) \sin \left (\alpha \right )d \alpha \right ) \operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )+\left (\int _{0}^{x}\operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i \alpha ^{\frac {3}{4}}}{3}\right ) \sin \left (\alpha \right )d \alpha \right ) \operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )\right )}{3} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )\right ) + \left (\frac {2 \pi \sqrt {x}\, \left (-\left (\int _{0}^{x}\operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i \alpha ^{\frac {3}{4}}}{3}\right ) \sin \left (\alpha \right )d \alpha \right ) \operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )+\left (\int _{0}^{x}\operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i \alpha ^{\frac {3}{4}}}{3}\right ) \sin \left (\alpha \right )d \alpha \right ) \operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )\right )}{3}\right ) \\ \end{align*} Initial conditions are used to solve for the constants of integration.

Looking at the above solution \begin {align*} y = c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )+\frac {2 \pi \sqrt {x}\, \left (-\left (\int _{0}^{x}\operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i \alpha ^{\frac {3}{4}}}{3}\right ) \sin \left (\alpha \right )d \alpha \right ) \operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )+\left (\int _{0}^{x}\operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i \alpha ^{\frac {3}{4}}}{3}\right ) \sin \left (\alpha \right )d \alpha \right ) \operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )\right )}{3} \tag {1} \end {align*}

Initial conditions are now substituted in the above solution. This will generate the required equations to solve for the integration constants. substituting \(y = 3\) and \(x = -2\) in the above gives \begin {align*} 3 = \frac {2 i \sqrt {2}\, \left (\pi \operatorname {BesselJ}\left (\frac {2}{3}, \left (-\frac {8}{3}-\frac {8 i}{3}\right ) 2^{\frac {1}{4}}\right ) \left (\int _{-2}^{0}\operatorname {BesselY}\left (\frac {2}{3}, \left (-\frac {4}{3}-\frac {4 i}{3}\right ) \left (-\alpha \right )^{\frac {3}{4}} \sqrt {2}\right ) \sin \left (\alpha \right )d \alpha \right )-\pi \operatorname {BesselY}\left (\frac {2}{3}, \left (-\frac {8}{3}-\frac {8 i}{3}\right ) 2^{\frac {1}{4}}\right ) \left (\int _{-2}^{0}\operatorname {BesselJ}\left (\frac {2}{3}, \left (-\frac {4}{3}-\frac {4 i}{3}\right ) \left (-\alpha \right )^{\frac {3}{4}} \sqrt {2}\right ) \sin \left (\alpha \right )d \alpha \right )+\frac {3 c_{1} \operatorname {BesselJ}\left (\frac {2}{3}, \left (-\frac {8}{3}-\frac {8 i}{3}\right ) 2^{\frac {1}{4}}\right )}{2}+\frac {3 c_{2} \operatorname {BesselY}\left (\frac {2}{3}, \left (-\frac {8}{3}-\frac {8 i}{3}\right ) 2^{\frac {1}{4}}\right )}{2}\right )}{3}\tag {1A} \end {align*}

Taking derivative of the solution gives \begin {align*} y^{\prime } = \frac {c_{1} \operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )}{2 \sqrt {x}}+2 i c_{1} x^{\frac {1}{4}} \left (\operatorname {BesselJ}\left (-\frac {1}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )+\frac {i \operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )}{4 x^{\frac {3}{4}}}\right )+\frac {c_{2} \operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )}{2 \sqrt {x}}+2 i c_{2} x^{\frac {1}{4}} \left (\operatorname {BesselY}\left (-\frac {1}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )+\frac {i \operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )}{4 x^{\frac {3}{4}}}\right )+\frac {\pi \left (-\left (\int _{0}^{x}\operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i \alpha ^{\frac {3}{4}}}{3}\right ) \sin \left (\alpha \right )d \alpha \right ) \operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )+\left (\int _{0}^{x}\operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i \alpha ^{\frac {3}{4}}}{3}\right ) \sin \left (\alpha \right )d \alpha \right ) \operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )\right )}{3 \sqrt {x}}+\frac {2 \pi \sqrt {x}\, \left (-\frac {2 i \left (\int _{0}^{x}\operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i \alpha ^{\frac {3}{4}}}{3}\right ) \sin \left (\alpha \right )d \alpha \right ) \left (\operatorname {BesselJ}\left (-\frac {1}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )+\frac {i \operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )}{4 x^{\frac {3}{4}}}\right )}{x^{\frac {1}{4}}}+\frac {2 i \left (\int _{0}^{x}\operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i \alpha ^{\frac {3}{4}}}{3}\right ) \sin \left (\alpha \right )d \alpha \right ) \left (\operatorname {BesselY}\left (-\frac {1}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )+\frac {i \operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )}{4 x^{\frac {3}{4}}}\right )}{x^{\frac {1}{4}}}\right )}{3} \end {align*}

substituting \(y^{\prime } = -1\) and \(x = -2\) in the above gives \begin {align*} -1 = \left (-\frac {2}{3}+\frac {2 i}{3}\right ) 2^{\frac {3}{4}} \left (\pi \left (\int _{-2}^{0}\operatorname {BesselY}\left (\frac {2}{3}, \left (-\frac {4}{3}-\frac {4 i}{3}\right ) \left (-\alpha \right )^{\frac {3}{4}} \sqrt {2}\right ) \sin \left (\alpha \right )d \alpha \right ) \operatorname {BesselJ}\left (-\frac {1}{3}, \left (-\frac {8}{3}-\frac {8 i}{3}\right ) 2^{\frac {1}{4}}\right )-\pi \left (\int _{-2}^{0}\operatorname {BesselJ}\left (\frac {2}{3}, \left (-\frac {4}{3}-\frac {4 i}{3}\right ) \left (-\alpha \right )^{\frac {3}{4}} \sqrt {2}\right ) \sin \left (\alpha \right )d \alpha \right ) \operatorname {BesselY}\left (-\frac {1}{3}, \left (-\frac {8}{3}-\frac {8 i}{3}\right ) 2^{\frac {1}{4}}\right )+\frac {3 c_{1} \operatorname {BesselJ}\left (-\frac {1}{3}, \left (-\frac {8}{3}-\frac {8 i}{3}\right ) 2^{\frac {1}{4}}\right )}{2}+\frac {3 c_{2} \operatorname {BesselY}\left (-\frac {1}{3}, \left (-\frac {8}{3}-\frac {8 i}{3}\right ) 2^{\frac {1}{4}}\right )}{2}\right )\tag {2A} \end {align*}

Equations {1A,2A} are now solved for \(\{c_{1}, c_{2}\}\). Solving for the constants gives \begin {align*} c_{1}&=\frac {2 \left (3 i 2^{\frac {3}{4}}-3 \,2^{\frac {3}{4}}\right ) \pi \operatorname {BesselY}\left (-\frac {1}{3}, \left (-\frac {8}{3}-\frac {8 i}{3}\right ) 2^{\frac {1}{4}}\right )}{3}+\frac {2 i \pi \sqrt {2}\, \operatorname {BesselY}\left (\frac {2}{3}, \left (-\frac {8}{3}-\frac {8 i}{3}\right ) 2^{\frac {1}{4}}\right )}{3}-\frac {2 \left (\int _{-2}^{0}\operatorname {BesselY}\left (\frac {2}{3}, \left (-\frac {4}{3}-\frac {4 i}{3}\right ) \left (-\alpha \right )^{\frac {3}{4}} \sqrt {2}\right ) \sin \left (\alpha \right )d \alpha \right ) \pi }{3}\\ c_{2}&=\frac {2 \left (-3 i 2^{\frac {3}{4}}+3 \,2^{\frac {3}{4}}\right ) \pi \operatorname {BesselJ}\left (-\frac {1}{3}, \left (-\frac {8}{3}-\frac {8 i}{3}\right ) 2^{\frac {1}{4}}\right )}{3}-\frac {2 i \pi \sqrt {2}\, \operatorname {BesselJ}\left (\frac {2}{3}, \left (-\frac {8}{3}-\frac {8 i}{3}\right ) 2^{\frac {1}{4}}\right )}{3}+\frac {2 \pi \left (\int _{-2}^{0}\operatorname {BesselJ}\left (\frac {2}{3}, \left (-\frac {4}{3}-\frac {4 i}{3}\right ) \left (-\alpha \right )^{\frac {3}{4}} \sqrt {2}\right ) \sin \left (\alpha \right )d \alpha \right )}{3} \end {align*}

Substituting these values back in above solution results in \begin {align*} y = -2 i \operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right ) 2^{\frac {3}{4}} \pi \operatorname {BesselJ}\left (-\frac {1}{3}, \left (-\frac {8}{3}-\frac {8 i}{3}\right ) 2^{\frac {1}{4}}\right ) \sqrt {x}+2 i \sqrt {x}\, \operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right ) 2^{\frac {3}{4}} \pi \operatorname {BesselY}\left (-\frac {1}{3}, \left (-\frac {8}{3}-\frac {8 i}{3}\right ) 2^{\frac {1}{4}}\right )+2 \operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right ) 2^{\frac {3}{4}} \pi \operatorname {BesselJ}\left (-\frac {1}{3}, \left (-\frac {8}{3}-\frac {8 i}{3}\right ) 2^{\frac {1}{4}}\right ) \sqrt {x}-2 \sqrt {x}\, \operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right ) 2^{\frac {3}{4}} \pi \operatorname {BesselY}\left (-\frac {1}{3}, \left (-\frac {8}{3}-\frac {8 i}{3}\right ) 2^{\frac {1}{4}}\right )-\frac {2 i \operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right ) \pi \sqrt {2}\, \operatorname {BesselJ}\left (\frac {2}{3}, \left (-\frac {8}{3}-\frac {8 i}{3}\right ) 2^{\frac {1}{4}}\right ) \sqrt {x}}{3}+\frac {2 i \sqrt {x}\, \operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right ) \sqrt {2}\, \operatorname {BesselY}\left (\frac {2}{3}, \left (-\frac {8}{3}-\frac {8 i}{3}\right ) 2^{\frac {1}{4}}\right ) \pi }{3}+\frac {2 \left (\int _{-2}^{0}\operatorname {BesselJ}\left (\frac {2}{3}, \left (-\frac {4}{3}-\frac {4 i}{3}\right ) \left (-\alpha \right )^{\frac {3}{4}} \sqrt {2}\right ) \sin \left (\alpha \right )d \alpha \right ) \pi \operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right ) \sqrt {x}}{3}-\frac {2 \left (\int _{-2}^{0}\operatorname {BesselY}\left (\frac {2}{3}, \left (-\frac {4}{3}-\frac {4 i}{3}\right ) \left (-\alpha \right )^{\frac {3}{4}} \sqrt {2}\right ) \sin \left (\alpha \right )d \alpha \right ) \sqrt {x}\, \operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right ) \pi }{3}-\frac {2 \pi \left (\int _{0}^{x}\operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i \alpha ^{\frac {3}{4}}}{3}\right ) \sin \left (\alpha \right )d \alpha \right ) \sqrt {x}\, \operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )}{3}+\frac {2 \pi \left (\int _{0}^{x}\operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i \alpha ^{\frac {3}{4}}}{3}\right ) \sin \left (\alpha \right )d \alpha \right ) \sqrt {x}\, \operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )}{3} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= 2 \sqrt {x}\, \left (\frac {\operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right ) \left (\int _{-2}^{0}\operatorname {BesselJ}\left (\frac {2}{3}, \left (-\frac {4}{3}-\frac {4 i}{3}\right ) \left (-\alpha \right )^{\frac {3}{4}} \sqrt {2}\right ) \sin \left (\alpha \right )d \alpha \right )}{3}-\frac {\operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right ) \left (\int _{-2}^{0}\operatorname {BesselY}\left (\frac {2}{3}, \left (-\frac {4}{3}-\frac {4 i}{3}\right ) \left (-\alpha \right )^{\frac {3}{4}} \sqrt {2}\right ) \sin \left (\alpha \right )d \alpha \right )}{3}+\frac {\left (\int _{0}^{x}\operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i \alpha ^{\frac {3}{4}}}{3}\right ) \sin \left (\alpha \right )d \alpha \right ) \operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )}{3}-\frac {\left (\int _{0}^{x}\operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i \alpha ^{\frac {3}{4}}}{3}\right ) \sin \left (\alpha \right )d \alpha \right ) \operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )}{3}+\left (\left (-1+i\right ) 2^{\frac {3}{4}} \operatorname {BesselY}\left (-\frac {1}{3}, \left (-\frac {8}{3}-\frac {8 i}{3}\right ) 2^{\frac {1}{4}}\right )+\frac {i \sqrt {2}\, \operatorname {BesselY}\left (\frac {2}{3}, \left (-\frac {8}{3}-\frac {8 i}{3}\right ) 2^{\frac {1}{4}}\right )}{3}\right ) \operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )-\left (\left (-1+i\right ) 2^{\frac {3}{4}} \operatorname {BesselJ}\left (-\frac {1}{3}, \left (-\frac {8}{3}-\frac {8 i}{3}\right ) 2^{\frac {1}{4}}\right )+\frac {i \sqrt {2}\, \operatorname {BesselJ}\left (\frac {2}{3}, \left (-\frac {8}{3}-\frac {8 i}{3}\right ) 2^{\frac {1}{4}}\right )}{3}\right ) \operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )\right ) \pi \\ \end{align*}

Veriﬁcation of solutions

\[ y = 2 \sqrt {x}\, \left (\frac {\operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right ) \left (\int _{-2}^{0}\operatorname {BesselJ}\left (\frac {2}{3}, \left (-\frac {4}{3}-\frac {4 i}{3}\right ) \left (-\alpha \right )^{\frac {3}{4}} \sqrt {2}\right ) \sin \left (\alpha \right )d \alpha \right )}{3}-\frac {\operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right ) \left (\int _{-2}^{0}\operatorname {BesselY}\left (\frac {2}{3}, \left (-\frac {4}{3}-\frac {4 i}{3}\right ) \left (-\alpha \right )^{\frac {3}{4}} \sqrt {2}\right ) \sin \left (\alpha \right )d \alpha \right )}{3}+\frac {\left (\int _{0}^{x}\operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i \alpha ^{\frac {3}{4}}}{3}\right ) \sin \left (\alpha \right )d \alpha \right ) \operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )}{3}-\frac {\left (\int _{0}^{x}\operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i \alpha ^{\frac {3}{4}}}{3}\right ) \sin \left (\alpha \right )d \alpha \right ) \operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )}{3}+\left (\left (-1+i\right ) 2^{\frac {3}{4}} \operatorname {BesselY}\left (-\frac {1}{3}, \left (-\frac {8}{3}-\frac {8 i}{3}\right ) 2^{\frac {1}{4}}\right )+\frac {i \sqrt {2}\, \operatorname {BesselY}\left (\frac {2}{3}, \left (-\frac {8}{3}-\frac {8 i}{3}\right ) 2^{\frac {1}{4}}\right )}{3}\right ) \operatorname {BesselJ}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )-\left (\left (-1+i\right ) 2^{\frac {3}{4}} \operatorname {BesselJ}\left (-\frac {1}{3}, \left (-\frac {8}{3}-\frac {8 i}{3}\right ) 2^{\frac {1}{4}}\right )+\frac {i \sqrt {2}\, \operatorname {BesselJ}\left (\frac {2}{3}, \left (-\frac {8}{3}-\frac {8 i}{3}\right ) 2^{\frac {1}{4}}\right )}{3}\right ) \operatorname {BesselY}\left (\frac {2}{3}, \frac {8 i x^{\frac {3}{4}}}{3}\right )\right ) \pi \] Veriﬁed OK.

Maple trace

`Methods for second order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying high order exact linear fully integrable 
trying differential order: 2; linear nonhomogeneous with symmetry [0,1] 
trying a double symmetry of the form [xi=0, eta=F(x)] 
-> Try solving first the homogeneous part of the ODE 
   trying a symmetry of the form [xi=0, eta=F(x)] 
   checking if the LODE is missing y 
   -> Trying an equivalence, under non-integer power transformations, 
      to LODEs admitting Liouvillian solutions. 
      -> Trying a Liouvillian solution using Kovacics algorithm 
      <- No Liouvillian solutions exists 
   -> Trying a solution in terms of special functions: 
      -> Bessel 
      -> elliptic 
      -> Legendre 
      -> Whittaker 
         -> hyper3: Equivalence to 1F1 under a power @ Moebius 
         <- hyper3 successful: received ODE is equivalent to the 0F1 ODE 
      <- Whittaker successful 
   <- special function solution successful 
<- solving first the homogeneous part of the ODE successful`

\[ y \left (x \right ) = \frac {4 \left (\left (\left (1-x \right )^{\frac {3}{2}}\right )^{\frac {2}{3}} \left (\left (\int _{-2}^{x}\frac {\operatorname {BesselI}\left (\frac {2}{3}, \frac {8 \sqrt {\left (-\textit {\_z1} +1\right )^{\frac {3}{2}}}}{3}\right ) \sqrt {-\textit {\_z1} +1}\, \sin \left (\textit {\_z1} \right )}{\left (\left (-\textit {\_z1} +1\right )^{\frac {3}{2}}\right )^{\frac {1}{3}}}d \textit {\_z1} \right ) \sqrt {3}+6 \,3^{\frac {3}{4}} \operatorname {BesselI}\left (-\frac {1}{3}, \frac {8 \,3^{\frac {3}{4}}}{3}\right )-3 \operatorname {BesselI}\left (\frac {2}{3}, \frac {8 \,3^{\frac {3}{4}}}{3}\right )\right ) \operatorname {BesselI}\left (-\frac {2}{3}, \frac {8 \sqrt {\left (1-x \right )^{\frac {3}{2}}}}{3}\right )+\left (-1+x \right ) \operatorname {BesselI}\left (\frac {2}{3}, \frac {8 \sqrt {\left (1-x \right )^{\frac {3}{2}}}}{3}\right ) \left (6 \,3^{\frac {3}{4}} \operatorname {BesselI}\left (\frac {1}{3}, \frac {8 \,3^{\frac {3}{4}}}{3}\right )+\left (\int _{-2}^{x}\frac {\operatorname {BesselI}\left (-\frac {2}{3}, \frac {8 \sqrt {\left (-\textit {\_z1} +1\right )^{\frac {3}{2}}}}{3}\right ) \left (\left (-\textit {\_z1} +1\right )^{\frac {3}{2}}\right )^{\frac {1}{3}} \sin \left (\textit {\_z1} \right )}{\sqrt {-\textit {\_z1} +1}}d \textit {\_z1} \right ) \sqrt {3}-3 \operatorname {BesselI}\left (-\frac {2}{3}, \frac {8 \,3^{\frac {3}{4}}}{3}\right )\right )\right ) \pi }{9 \left (\left (1-x \right )^{\frac {3}{2}}\right )^{\frac {1}{3}}} \]

9.5 problem 5

9.5.1 Existence and uniqueness analysis

9.5.2 Solving as second order bessel ode ode