Internal problem ID [11299]
Internal file name [OUTPUT/10285_Wednesday_December_21_2022_03_47_42_PM_71014606/index.tex]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter VIII, Linear differential equations of the second order. Article 54. Change of
independent variable. Page 127
Problem number: Ex 4.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "kovacic", "second_order_bessel_ode", "second_order_change_of_variable_on_x_method_1", "second_order_change_of_variable_on_x_method_2"
Maple gives the following as the ode type
[[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {x^{6} y^{\prime \prime }+3 x^{5} y^{\prime }+y=\frac {1}{x^{2}}} \]
This is second order non-homogeneous ODE. Let the solution be \[ y = y_h + y_p \] Where \(y_h\) is the solution to the homogeneous ODE \( A y''(x) + B y'(x) + C y(x) = 0\), and \(y_p\) is a particular solution to the non-homogeneous ODE \(A y''(x) + B y'(x) + C y(x) = f(x)\). \(y_h\) is the solution to \[ x^{6} y^{\prime \prime }+3 x^{5} y^{\prime }+y = 0 \] In normal form the ode \begin {align*} x^{6} y^{\prime \prime }+3 x^{5} y^{\prime }+y&=0 \tag {1} \end {align*}
Becomes \begin {align*} y^{\prime \prime }+p \left (x \right ) y^{\prime }+q \left (x \right ) y&=0 \tag {2} \end {align*}
Where \begin {align*} p \left (x \right )&=\frac {3}{x}\\ q \left (x \right )&=\frac {1}{x^{6}} \end {align*}
Applying change of variables \(\tau = g \left (x \right )\) to (2) gives \begin {align*} \frac {d^{2}}{d \tau ^{2}}y \left (\tau \right )+p_{1} \left (\frac {d}{d \tau }y \left (\tau \right )\right )+q_{1} y \left (\tau \right )&=0 \tag {3} \end {align*}
Where \(\tau \) is the new independent variable, and \begin {align*} p_{1} \left (\tau \right ) &=\frac {\tau ^{\prime \prime }\left (x \right )+p \left (x \right ) \tau ^{\prime }\left (x \right )}{{\tau ^{\prime }\left (x \right )}^{2}}\tag {4} \\ q_{1} \left (\tau \right ) &=\frac {q \left (x \right )}{{\tau ^{\prime }\left (x \right )}^{2}}\tag {5} \end {align*}
Let \(p_{1} = 0\). Eq (4) simplifies to \begin {align*} \tau ^{\prime \prime }\left (x \right )+p \left (x \right ) \tau ^{\prime }\left (x \right )&=0 \end {align*}
This ode is solved resulting in \begin {align*} \tau &= \int {\mathrm e}^{-\left (\int p \left (x \right )d x \right )}d x\\ &= \int {\mathrm e}^{-\left (\int \frac {3}{x}d x \right )}d x\\ &= \int e^{-3 \ln \left (x \right )} \,dx\\ &= \int \frac {1}{x^{3}}d x\\ &= -\frac {1}{2 x^{2}}\tag {6} \end {align*}
Using (6) to evaluate \(q_{1}\) from (5) gives \begin {align*} q_{1} \left (\tau \right ) &= \frac {q \left (x \right )}{{\tau ^{\prime }\left (x \right )}^{2}}\\ &= \frac {\frac {1}{x^{6}}}{\frac {1}{x^{6}}}\\ &= 1\tag {7} \end {align*}
Substituting the above in (3) and noting that now \(p_{1} = 0\) results in \begin {align*} \frac {d^{2}}{d \tau ^{2}}y \left (\tau \right )+q_{1} y \left (\tau \right )&=0 \\ \frac {d^{2}}{d \tau ^{2}}y \left (\tau \right )+y \left (\tau \right )&=0 \end {align*}
The above ode is now solved for \(y \left (\tau \right )\).This is second order with constant coefficients homogeneous ODE. In standard form the ODE is \[ A y''(\tau ) + B y'(\tau ) + C y(\tau ) = 0 \] Where in the above \(A=1, B=0, C=1\). Let the solution be \(y \left (\tau \right )=e^{\lambda \tau }\). Substituting this into the ODE gives \[ \lambda ^{2} {\mathrm e}^{\lambda \tau }+{\mathrm e}^{\lambda \tau } = 0 \tag {1} \] Since exponential function is never zero, then dividing Eq(2) throughout by \(e^{\lambda \tau }\) gives \[ \lambda ^{2}+1 = 0 \tag {2} \] Equation (2) is the characteristic equation of the ODE. Its roots determine the general solution form.Using the quadratic formula \[ \lambda _{1,2} = \frac {-B}{2 A} \pm \frac {1}{2 A} \sqrt {B^2 - 4 A C} \] Substituting \(A=1, B=0, C=1\) into the above gives \begin {align*} \lambda _{1,2} &= \frac {0}{(2) \left (1\right )} \pm \frac {1}{(2) \left (1\right )} \sqrt {0^2 - (4) \left (1\right )\left (1\right )}\\ &= \pm i \end {align*}
Hence \begin {align*} \lambda _1 &= + i\\ \lambda _2 &= - i \end {align*}
Which simplifies to \begin{align*} \lambda _1 &= i \\ \lambda _2 &= -i \\ \end{align*} Since roots are complex conjugate of each others, then let the roots be \[ \lambda _{1,2} = \alpha \pm i \beta \] Where \(\alpha =0\) and \(\beta =1\). Therefore the final solution, when using Euler relation, can be written as \[ y \left (\tau \right ) = e^{\alpha \tau } \left ( c_{1} \cos (\beta \tau ) + c_{2} \sin (\beta \tau ) \right ) \] Which becomes \[ y \left (\tau \right ) = e^{0}\left (c_{1} \cos \left (\tau \right )+c_{2} \sin \left (\tau \right )\right ) \] Or \[ y \left (\tau \right ) = c_{1} \cos \left (\tau \right )+c_{2} \sin \left (\tau \right ) \] The above solution is now transformed back to \(y\) using (6) which results in \begin {align*} y &= c_{1} \cos \left (\frac {1}{2 x^{2}}\right )-c_{2} \sin \left (\frac {1}{2 x^{2}}\right ) \end {align*}
Therefore the homogeneous solution \(y_h\) is \[ y_h = c_{1} \cos \left (\frac {1}{2 x^{2}}\right )-c_{2} \sin \left (\frac {1}{2 x^{2}}\right ) \] 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 &= \cos \left (\frac {1}{2 x^{2}}\right ) \\ y_2 &= \sin \left (\frac {1}{2 x^{2}}\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 \[ W = \begin {vmatrix} \cos \left (\frac {1}{2 x^{2}}\right ) & \sin \left (\frac {1}{2 x^{2}}\right ) \\ \frac {d}{dx}\left (\cos \left (\frac {1}{2 x^{2}}\right )\right ) & \frac {d}{dx}\left (\sin \left (\frac {1}{2 x^{2}}\right )\right ) \end {vmatrix} \] Which gives \[ W = \begin {vmatrix} \cos \left (\frac {1}{2 x^{2}}\right ) & \sin \left (\frac {1}{2 x^{2}}\right ) \\ \frac {\sin \left (\frac {1}{2 x^{2}}\right )}{x^{3}} & -\frac {\cos \left (\frac {1}{2 x^{2}}\right )}{x^{3}} \end {vmatrix} \] Therefore \[ W = \left (\cos \left (\frac {1}{2 x^{2}}\right )\right )\left (-\frac {\cos \left (\frac {1}{2 x^{2}}\right )}{x^{3}}\right ) - \left (\sin \left (\frac {1}{2 x^{2}}\right )\right )\left (\frac {\sin \left (\frac {1}{2 x^{2}}\right )}{x^{3}}\right ) \] Which simplifies to \[ W = -\frac {\cos \left (\frac {1}{2 x^{2}}\right )^{2}+\sin \left (\frac {1}{2 x^{2}}\right )^{2}}{x^{3}} \] Which simplifies to \[ W = -\frac {1}{x^{3}} \] Therefore Eq. (2) becomes \[ u_1 = -\int \frac {\frac {\sin \left (\frac {1}{2 x^{2}}\right )}{x^{2}}}{-x^{3}}\,dx \] Which simplifies to \[ u_1 = - \int -\frac {\sin \left (\frac {1}{2 x^{2}}\right )}{x^{5}}d x \] Hence \[ u_1 = \frac {\cos \left (\frac {1}{2 x^{2}}\right )}{x^{2}}-2 \sin \left (\frac {1}{2 x^{2}}\right ) \] And Eq. (3) becomes \[ u_2 = \int \frac {\frac {\cos \left (\frac {1}{2 x^{2}}\right )}{x^{2}}}{-x^{3}}\,dx \] Which simplifies to \[ u_2 = \int -\frac {\cos \left (\frac {1}{2 x^{2}}\right )}{x^{5}}d x \] Hence \[ u_2 = \frac {\sin \left (\frac {1}{2 x^{2}}\right )}{x^{2}}+2 \cos \left (\frac {1}{2 x^{2}}\right ) \] Therefore the particular solution, from equation (1) is \[ y_p(x) = \left (\frac {\cos \left (\frac {1}{2 x^{2}}\right )}{x^{2}}-2 \sin \left (\frac {1}{2 x^{2}}\right )\right ) \cos \left (\frac {1}{2 x^{2}}\right )+\left (\frac {\sin \left (\frac {1}{2 x^{2}}\right )}{x^{2}}+2 \cos \left (\frac {1}{2 x^{2}}\right )\right ) \sin \left (\frac {1}{2 x^{2}}\right ) \] Which simplifies to \[ y_p(x) = \frac {1}{x^{2}} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (c_{1} \cos \left (\frac {1}{2 x^{2}}\right )-c_{2} \sin \left (\frac {1}{2 x^{2}}\right )\right ) + \left (\frac {1}{x^{2}}\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} \cos \left (\frac {1}{2 x^{2}}\right )-c_{2} \sin \left (\frac {1}{2 x^{2}}\right )+\frac {1}{x^{2}} \\ \end{align*}
Verification of solutions
\[ y = c_{1} \cos \left (\frac {1}{2 x^{2}}\right )-c_{2} \sin \left (\frac {1}{2 x^{2}}\right )+\frac {1}{x^{2}} \] Verified OK.
This is second order non-homogeneous ODE. In standard form the ODE is \[ A y''(x) + B y'(x) + C y(x) = f(x) \] Where \(A=x^{6}, B=3 x^{5}, C=1, f(x)=\frac {1}{x^{2}}\). Let the solution be \[ y = y_h + y_p \] Where \(y_h\) is the solution to the homogeneous ODE \( A y''(x) + B y'(x) + C y(x) = 0\), and \(y_p\) is a particular solution to the non-homogeneous ODE \(A y''(x) + B y'(x) + C y(x) = f(x)\). Solving for \(y_h\) from \[ x^{6} y^{\prime \prime }+3 x^{5} y^{\prime }+y = 0 \] In normal form the ode \begin {align*} x^{6} y^{\prime \prime }+3 x^{5} y^{\prime }+y&=0 \tag {1} \end {align*}
Becomes \begin {align*} y^{\prime \prime }+p \left (x \right ) y^{\prime }+q \left (x \right ) y&=0 \tag {2} \end {align*}
Where \begin {align*} p \left (x \right )&=\frac {3}{x}\\ q \left (x \right )&=\frac {1}{x^{6}} \end {align*}
Applying change of variables \(\tau = g \left (x \right )\) to (2) results \begin {align*} \frac {d^{2}}{d \tau ^{2}}y \left (\tau \right )+p_{1} \left (\frac {d}{d \tau }y \left (\tau \right )\right )+q_{1} y \left (\tau \right )&=0 \tag {3} \end {align*}
Where \(\tau \) is the new independent variable, and \begin {align*} p_{1} \left (\tau \right ) &=\frac {\tau ^{\prime \prime }\left (x \right )+p \left (x \right ) \tau ^{\prime }\left (x \right )}{{\tau ^{\prime }\left (x \right )}^{2}}\tag {4} \\ q_{1} \left (\tau \right ) &=\frac {q \left (x \right )}{{\tau ^{\prime }\left (x \right )}^{2}}\tag {5} \end {align*}
Let \(q_1=c^2\) where \(c\) is some constant. Therefore from (5) \begin {align*} \tau ' &= \frac {1}{c}\sqrt {q}\\ &= \frac {\sqrt {\frac {1}{x^{6}}}}{c}\tag {6} \\ \tau '' &= -\frac {3}{c \sqrt {\frac {1}{x^{6}}}\, x^{7}} \end {align*}
Substituting the above into (4) results in \begin {align*} p_{1} \left (\tau \right ) &=\frac {\tau ^{\prime \prime }\left (x \right )+p \left (x \right ) \tau ^{\prime }\left (x \right )}{{\tau ^{\prime }\left (x \right )}^{2}}\\ &=\frac {-\frac {3}{c \sqrt {\frac {1}{x^{6}}}\, x^{7}}+\frac {3}{x}\frac {\sqrt {\frac {1}{x^{6}}}}{c}}{\left (\frac {\sqrt {\frac {1}{x^{6}}}}{c}\right )^2} \\ &=0 \end {align*}
Therefore ode (3) now becomes \begin {align*} y \left (\tau \right )'' + p_1 y \left (\tau \right )' + q_1 y \left (\tau \right ) &= 0 \\ \frac {d^{2}}{d \tau ^{2}}y \left (\tau \right )+c^{2} y \left (\tau \right ) &= 0 \tag {7} \end {align*}
The above ode is now solved for \(y \left (\tau \right )\). Since the ode is now constant coefficients, it can be easily solved to give \begin {align*} y \left (\tau \right ) &= c_{1} \cos \left (c \tau \right )+c_{2} \sin \left (c \tau \right ) \end {align*}
Now from (6) \begin {align*} \tau &= \int \frac {1}{c} \sqrt q \,dx \\ &= \frac {\int \sqrt {\frac {1}{x^{6}}}d x}{c}\\ &= -\frac {x \sqrt {\frac {1}{x^{6}}}}{2 c} \end {align*}
Substituting the above into the solution obtained gives \[ y = c_{1} \cos \left (\frac {1}{2 x^{2}}\right )-c_{2} \sin \left (\frac {1}{2 x^{2}}\right ) \] Now the particular solution to this ODE is found \[ x^{6} y^{\prime \prime }+3 x^{5} y^{\prime }+y = \frac {1}{x^{2}} \] 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 &= \cos \left (\frac {1}{2 x^{2}}\right ) \\ y_2 &= \sin \left (\frac {1}{2 x^{2}}\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 \[ W = \begin {vmatrix} \cos \left (\frac {1}{2 x^{2}}\right ) & \sin \left (\frac {1}{2 x^{2}}\right ) \\ \frac {d}{dx}\left (\cos \left (\frac {1}{2 x^{2}}\right )\right ) & \frac {d}{dx}\left (\sin \left (\frac {1}{2 x^{2}}\right )\right ) \end {vmatrix} \] Which gives \[ W = \begin {vmatrix} \cos \left (\frac {1}{2 x^{2}}\right ) & \sin \left (\frac {1}{2 x^{2}}\right ) \\ \frac {\sin \left (\frac {1}{2 x^{2}}\right )}{x^{3}} & -\frac {\cos \left (\frac {1}{2 x^{2}}\right )}{x^{3}} \end {vmatrix} \] Therefore \[ W = \left (\cos \left (\frac {1}{2 x^{2}}\right )\right )\left (-\frac {\cos \left (\frac {1}{2 x^{2}}\right )}{x^{3}}\right ) - \left (\sin \left (\frac {1}{2 x^{2}}\right )\right )\left (\frac {\sin \left (\frac {1}{2 x^{2}}\right )}{x^{3}}\right ) \] Which simplifies to \[ W = -\frac {\cos \left (\frac {1}{2 x^{2}}\right )^{2}+\sin \left (\frac {1}{2 x^{2}}\right )^{2}}{x^{3}} \] Which simplifies to \[ W = -\frac {1}{x^{3}} \] Therefore Eq. (2) becomes \[ u_1 = -\int \frac {\frac {\sin \left (\frac {1}{2 x^{2}}\right )}{x^{2}}}{-x^{3}}\,dx \] Which simplifies to \[ u_1 = - \int -\frac {\sin \left (\frac {1}{2 x^{2}}\right )}{x^{5}}d x \] Hence \[ u_1 = \frac {\cos \left (\frac {1}{2 x^{2}}\right )}{x^{2}}-2 \sin \left (\frac {1}{2 x^{2}}\right ) \] And Eq. (3) becomes \[ u_2 = \int \frac {\frac {\cos \left (\frac {1}{2 x^{2}}\right )}{x^{2}}}{-x^{3}}\,dx \] Which simplifies to \[ u_2 = \int -\frac {\cos \left (\frac {1}{2 x^{2}}\right )}{x^{5}}d x \] Hence \[ u_2 = \frac {\sin \left (\frac {1}{2 x^{2}}\right )}{x^{2}}+2 \cos \left (\frac {1}{2 x^{2}}\right ) \] Therefore the particular solution, from equation (1) is \[ y_p(x) = \left (\frac {\cos \left (\frac {1}{2 x^{2}}\right )}{x^{2}}-2 \sin \left (\frac {1}{2 x^{2}}\right )\right ) \cos \left (\frac {1}{2 x^{2}}\right )+\left (\frac {\sin \left (\frac {1}{2 x^{2}}\right )}{x^{2}}+2 \cos \left (\frac {1}{2 x^{2}}\right )\right ) \sin \left (\frac {1}{2 x^{2}}\right ) \] Which simplifies to \[ y_p(x) = \frac {1}{x^{2}} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (c_{1} \cos \left (\frac {1}{2 x^{2}}\right )-c_{2} \sin \left (\frac {1}{2 x^{2}}\right )\right ) + \left (\frac {1}{x^{2}}\right ) \\ &= c_{1} \cos \left (\frac {1}{2 x^{2}}\right )-c_{2} \sin \left (\frac {1}{2 x^{2}}\right )+\frac {1}{x^{2}} \\ \end{align*} Which simplifies to \[ y = c_{1} \cos \left (\frac {1}{2 x^{2}}\right )-c_{2} \sin \left (\frac {1}{2 x^{2}}\right )+\frac {1}{x^{2}} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} \cos \left (\frac {1}{2 x^{2}}\right )-c_{2} \sin \left (\frac {1}{2 x^{2}}\right )+\frac {1}{x^{2}} \\ \end{align*}
Verification of solutions
\[ y = c_{1} \cos \left (\frac {1}{2 x^{2}}\right )-c_{2} \sin \left (\frac {1}{2 x^{2}}\right )+\frac {1}{x^{2}} \] Verified OK.
Writing the ode as \begin {align*} x^{2} y^{\prime \prime }+3 y^{\prime } x +\frac {y}{x^{4}} = \frac {1}{x^{6}}\tag {1} \end {align*}
Let the solution be \begin {align*} y &= y_h + y_p \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 &= -1\\ \beta &= {\frac {1}{2}}\\ n &= {\frac {1}{2}}\\ \gamma &= -2 \end {align*}
Substituting all the above into (4) gives the solution as \begin {align*} y = \frac {2 c_{1} \sin \left (\frac {1}{2 x^{2}}\right )}{x \sqrt {\pi }\, \sqrt {\frac {1}{x^{2}}}}-\frac {2 c_{2} \cos \left (\frac {1}{2 x^{2}}\right )}{x \sqrt {\pi }\, \sqrt {\frac {1}{x^{2}}}} \end {align*}
Therefore the homogeneous solution \(y_h\) is \[ y_h = \frac {2 c_{1} \sin \left (\frac {1}{2 x^{2}}\right )}{x \sqrt {\pi }\, \sqrt {\frac {1}{x^{2}}}}-\frac {2 c_{2} \cos \left (\frac {1}{2 x^{2}}\right )}{x \sqrt {\pi }\, \sqrt {\frac {1}{x^{2}}}} \] 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 &= \cos \left (\frac {1}{2 x^{2}}\right ) \\ y_2 &= \sin \left (\frac {1}{2 x^{2}}\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 \[ W = \begin {vmatrix} \cos \left (\frac {1}{2 x^{2}}\right ) & \sin \left (\frac {1}{2 x^{2}}\right ) \\ \frac {d}{dx}\left (\cos \left (\frac {1}{2 x^{2}}\right )\right ) & \frac {d}{dx}\left (\sin \left (\frac {1}{2 x^{2}}\right )\right ) \end {vmatrix} \] Which gives \[ W = \begin {vmatrix} \cos \left (\frac {1}{2 x^{2}}\right ) & \sin \left (\frac {1}{2 x^{2}}\right ) \\ \frac {\sin \left (\frac {1}{2 x^{2}}\right )}{x^{3}} & -\frac {\cos \left (\frac {1}{2 x^{2}}\right )}{x^{3}} \end {vmatrix} \] Therefore \[ W = \left (\cos \left (\frac {1}{2 x^{2}}\right )\right )\left (-\frac {\cos \left (\frac {1}{2 x^{2}}\right )}{x^{3}}\right ) - \left (\sin \left (\frac {1}{2 x^{2}}\right )\right )\left (\frac {\sin \left (\frac {1}{2 x^{2}}\right )}{x^{3}}\right ) \] Which simplifies to \[ W = -\frac {\cos \left (\frac {1}{2 x^{2}}\right )^{2}+\sin \left (\frac {1}{2 x^{2}}\right )^{2}}{x^{3}} \] Which simplifies to \[ W = -\frac {1}{x^{3}} \] Therefore Eq. (2) becomes \[ u_1 = -\int \frac {\frac {\sin \left (\frac {1}{2 x^{2}}\right )}{x^{6}}}{-\frac {1}{x}}\,dx \] Which simplifies to \[ u_1 = - \int -\frac {\sin \left (\frac {1}{2 x^{2}}\right )}{x^{5}}d x \] Hence \[ u_1 = \frac {\cos \left (\frac {1}{2 x^{2}}\right )}{x^{2}}-2 \sin \left (\frac {1}{2 x^{2}}\right ) \] And Eq. (3) becomes \[ u_2 = \int \frac {\frac {\cos \left (\frac {1}{2 x^{2}}\right )}{x^{6}}}{-\frac {1}{x}}\,dx \] Which simplifies to \[ u_2 = \int -\frac {\cos \left (\frac {1}{2 x^{2}}\right )}{x^{5}}d x \] Hence \[ u_2 = \frac {\sin \left (\frac {1}{2 x^{2}}\right )}{x^{2}}+2 \cos \left (\frac {1}{2 x^{2}}\right ) \] Therefore the particular solution, from equation (1) is \[ y_p(x) = \left (\frac {\cos \left (\frac {1}{2 x^{2}}\right )}{x^{2}}-2 \sin \left (\frac {1}{2 x^{2}}\right )\right ) \cos \left (\frac {1}{2 x^{2}}\right )+\left (\frac {\sin \left (\frac {1}{2 x^{2}}\right )}{x^{2}}+2 \cos \left (\frac {1}{2 x^{2}}\right )\right ) \sin \left (\frac {1}{2 x^{2}}\right ) \] Which simplifies to \[ y_p(x) = \frac {1}{x^{2}} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (\frac {2 c_{1} \sin \left (\frac {1}{2 x^{2}}\right )}{x \sqrt {\pi }\, \sqrt {\frac {1}{x^{2}}}}-\frac {2 c_{2} \cos \left (\frac {1}{2 x^{2}}\right )}{x \sqrt {\pi }\, \sqrt {\frac {1}{x^{2}}}}\right ) + \left (\frac {1}{x^{2}}\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {2 c_{1} \sin \left (\frac {1}{2 x^{2}}\right )}{x \sqrt {\pi }\, \sqrt {\frac {1}{x^{2}}}}-\frac {2 c_{2} \cos \left (\frac {1}{2 x^{2}}\right )}{x \sqrt {\pi }\, \sqrt {\frac {1}{x^{2}}}}+\frac {1}{x^{2}} \\ \end{align*}
Verification of solutions
\[ y = \frac {2 c_{1} \sin \left (\frac {1}{2 x^{2}}\right )}{x \sqrt {\pi }\, \sqrt {\frac {1}{x^{2}}}}-\frac {2 c_{2} \cos \left (\frac {1}{2 x^{2}}\right )}{x \sqrt {\pi }\, \sqrt {\frac {1}{x^{2}}}}+\frac {1}{x^{2}} \] Verified OK.
Writing the ode as \begin {align*} x^{6} y^{\prime \prime }+3 x^{5} y^{\prime }+y &= 0 \tag {1} \\ A y^{\prime \prime } + B y^{\prime } + C y &= 0 \tag {2} \end {align*}
Comparing (1) and (2) shows that \begin {align*} A &= x^{6} \\ B &= 3 x^{5}\tag {3} \\ C &= 1 \end {align*}
Applying the Liouville transformation on the dependent variable gives \begin {align*} z(x) &= y e^{\int \frac {B}{2 A} \,dx} \end {align*}
Then (2) becomes \begin {align*} z''(x) = r z(x)\tag {4} \end {align*}
Where \(r\) is given by \begin {align*} r &= \frac {s}{t}\tag {5} \\ &= \frac {2 A B' - 2 B A' + B^2 - 4 A C}{4 A^2} \end {align*}
Substituting the values of \(A,B,C\) from (3) in the above and simplifying gives \begin {align*} r &= \frac {3 x^{4}-4}{4 x^{6}}\tag {6} \end {align*}
Comparing the above to (5) shows that \begin {align*} s &= 3 x^{4}-4\\ t &= 4 x^{6} \end {align*}
Therefore eq. (4) becomes \begin {align*} z''(x) &= \left ( \frac {3 x^{4}-4}{4 x^{6}}\right ) z(x)\tag {7} \end {align*}
Equation (7) is now solved. After finding \(z(x)\) then \(y\) is found using the inverse transformation \begin {align*} y &= z \left (x \right ) e^{-\int \frac {B}{2 A} \,dx} \end {align*}
The first step is to determine the case of Kovacic algorithm this ode belongs to. There are 3 cases depending on the order of poles of \(r\) and the order of \(r\) at \(\infty \). The following table summarizes these cases.
| Case |
Allowed pole order for \(r\) |
Allowed value for \(\mathcal {O}(\infty )\) |
| 1 |
\(\left \{ 0,1,2,4,6,8,\cdots \right \} \) |
\(\left \{ \cdots ,-6,-4,-2,0,2,3,4,5,6,\cdots \right \} \) |
|
2 |
Need to have at least one pole that is either order \(2\) or odd order greater than \(2\). Any other pole order is allowed as long as the above condition is satisfied. Hence the following set of pole orders are all allowed. \(\{1,2\}\),\(\{1,3\}\),\(\{2\}\),\(\{3\}\),\(\{3,4\}\),\(\{1,2,5\}\). |
no condition |
| 3 |
\(\left \{ 1,2\right \} \) |
\(\left \{ 2,3,4,5,6,7,\cdots \right \} \) |
The order of \(r\) at \(\infty \) is the degree of \(t\) minus the degree of \(s\). Therefore \begin {align*} O\left (\infty \right ) &= \text {deg}(t) - \text {deg}(s) \\ &= 6 - 4 \\ &= 2 \end {align*}
The poles of \(r\) in eq. (7) and the order of each pole are determined by solving for the roots of \(t=4 x^{6}\). There is a pole at \(x=0\) of order \(6\). Since there is no odd order pole larger than \(2\) and the order at \(\infty \) is \(2\) then the necessary conditions for case one are met. Therefore \begin {align*} L &= [1] \end {align*}
Attempting to find a solution using case \(n=1\).
Looking at higher order poles of order \(2 v \)≥\( 4\) (must be even order for case one).Then for each pole \(c\), \([\sqrt r]_{c}\) is the sum of terms \(\frac {1}{(x-c)^i}\) for \(2 \leq i \leq v\) in the Laurent series expansion of \(\sqrt r\) expanded around each pole \(c\). Hence \begin {align*} [\sqrt r]_c &= \sum _2^v \frac {a_i}{ (x-c)^i} \tag {1B} \end {align*}
Let \(a\) be the coefficient of the term \(\frac {1}{ (x-c)^v}\) in the above where \(v\) is the pole order divided by 2. Let \(b\) be the coefficient of \(\frac {1}{ (x-c)^{v+1}} \) in \(r\) minus the coefficient of \(\frac {1}{ (x-c)^{v+1}} \) in \([\sqrt r]_c\). Then \begin {alignat*} {1} \alpha _c^{+} &= \frac {1}{2} \left ( \frac {b}{a} + v \right ) \\ \alpha _c^{-} &= \frac {1}{2} \left (- \frac {b}{a} + v \right ) \end {alignat*}
The partial fraction decomposition of \(r\) is \[ r = -\frac {1}{x^{6}}+\frac {3}{4 x^{2}} \] There is pole in \(r\) at \(x= 0\) of order \(6\), hence \(v=3\). Expanding \(\sqrt {r}\) as Laurent series about this pole \(c=0\) gives \[ [\sqrt {r}]_c \approx \frac {i}{x^{3}}-\frac {3 i x}{8} + \dots \tag {2B} \] Using eq. (1B), taking the sum up to \(v=3\) the above becomes \[ [\sqrt {r}]_c = \frac {i}{x^{3}} \tag {3B} \] The above shows that the coefficient of \(\frac {1}{(x-0)^{3}}\) is \[ a = i \] Now we need to find \(b\). let \(b\) be the coefficient of the term \(\frac {1}{(x-c)^{v+1}}\) in \(r\) minus the coefficient of the same term but in the sum \([\sqrt r]_c \) found in eq. (3B). Here \(c\) is current pole which is \(c=0\). This term becomes \(\frac {1}{x^{4}}\). The coefficient of this term in the sum \([\sqrt r]_c\) is seen to be \(0\) and the coefficient of this term \(r\) is found from the partial fraction decomposition from above to be \(0\). Therefore \begin {align*} b &= \left (0\right )-(0)\\ &= 0 \end {align*}
Hence \begin {alignat*} {3} [\sqrt r]_c &= \frac {i}{x^{3}} \\ \alpha _c^{+} &= \frac {1}{2} \left ( \frac {b}{a} + v \right ) &&= \frac {1}{2} \left ( \frac {0}{i} + 3 \right ) &&={\frac {3}{2}}\\ \alpha _c^{-} &= \frac {1}{2} \left (- \frac {b}{a} + v \right ) &&= \frac {1}{2} \left (- \frac {0}{i} + 3 \right )&&={\frac {3}{2}} \end {alignat*}
Since the order of \(r\) at \(\infty \) is 2 then \([\sqrt r]_\infty =0\). Let \(b\) be the coefficient of \(\frac {1}{x^{2}}\) in the Laurent series expansion of \(r\) at \(\infty \). which can be found by dividing the leading coefficient of \(s\) by the leading coefficient of \(t\) from \begin {alignat*} {2} r &= \frac {s}{t} &&= \frac {3 x^{4}-4}{4 x^{6}} \end {alignat*}
Since the \(\text {gcd}(s,t)=1\). This gives \(b={\frac {3}{4}}\). Hence \begin {alignat*} {2} [\sqrt r]_\infty &= 0 \\ \alpha _{\infty }^{+} &= \frac {1}{2} + \sqrt {1+4 b} &&= {\frac {3}{2}}\\ \alpha _{\infty }^{-} &= \frac {1}{2} - \sqrt {1+4 b} &&= -{\frac {1}{2}} \end {alignat*}
The following table summarizes the findings so far for poles and for the order of \(r\) at \(\infty \) where \(r\) is \[ r=\frac {3 x^{4}-4}{4 x^{6}} \]
| pole \(c\) location | pole order | \([\sqrt r]_c\) | \(\alpha _c^{+}\) | \(\alpha _c^{-}\) |
| \(0\) | \(6\) | \(\frac {i}{x^{3}}\) | \(\frac {3}{2}\) | \(\frac {3}{2}\) |
| Order of \(r\) at \(\infty \) | \([\sqrt r]_\infty \) | \(\alpha _\infty ^{+}\) | \(\alpha _\infty ^{-}\) |
| \(2\) | \(0\) | \(\frac {3}{2}\) | \(-{\frac {1}{2}}\) |
Now that the all \([\sqrt r]_c\) and its associated \(\alpha _c^{\pm }\) have been determined for all the poles in the set \(\Gamma \) and \([\sqrt r]_\infty \) and its associated \(\alpha _\infty ^{\pm }\) have also been found, the next step is to determine possible non negative integer \(d\) from these using \begin {align*} d &= \alpha _\infty ^{s(\infty )} - \sum _{c \in \Gamma } \alpha _c^{s(c)} \end {align*}
Where \(s(c)\) is either \(+\) or \(-\) and \(s(\infty )\) is the sign of \(\alpha _\infty ^{\pm }\). This is done by trial over all set of families \(s=(s(c))_{c \in \Gamma \cup {\infty }}\) until such \(d\) is found to work in finding candidate \(\omega \). Trying \(\alpha _\infty ^{+} = {\frac {3}{2}}\) then \begin {align*} d &= \alpha _\infty ^{+} - \left ( \alpha _{c_1}^{-} \right ) \\ &= {\frac {3}{2}} - \left ( {\frac {3}{2}} \right ) \\ &= 0 \end {align*}
Since \(d\) an integer and \(d \geq 0\) then it can be used to find \(\omega \) using \begin {align*} \omega &= \sum _{c \in \Gamma } \left ( s(c) [\sqrt r]_c + \frac {\alpha _c^{s(c)}}{x-c} \right ) + s(\infty ) [\sqrt r]_\infty \end {align*}
Substituting the above values in the above results in \begin {align*} \omega &= \left ( (-)[\sqrt r]_{c_1} + \frac { \alpha _{c_1}^{-} }{x- c_1}\right ) + (+) [\sqrt r]_\infty \\ &= -\frac {i}{x^{3}}+\frac {3}{2 x} + \left ( 0 \right ) \\ &= -\frac {i}{x^{3}}+\frac {3}{2 x}\\ &= -\frac {i}{x^{3}}+\frac {3}{2 x} \end {align*}
Now that \(\omega \) is determined, the next step is find a corresponding minimal polynomial \(p(x)\) of degree \(d=0\) to solve the ode. The polynomial \(p(x)\) needs to satisfy the equation \begin {align*} p'' + 2 \omega p' + \left ( \omega ' +\omega ^2 -r\right ) p = 0 \tag {1A} \end {align*}
Let \begin {align*} p(x) &= 1\tag {2A} \end {align*}
Substituting the above in eq. (1A) gives \begin {align*} \left (0\right ) + 2 \left (-\frac {i}{x^{3}}+\frac {3}{2 x}\right ) \left (0\right ) + \left ( \left (\frac {3 i}{x^{4}}-\frac {3}{2 x^{2}}\right ) + \left (-\frac {i}{x^{3}}+\frac {3}{2 x}\right )^2 - \left (\frac {3 x^{4}-4}{4 x^{6}}\right ) \right ) &= 0\\ 0 = 0 \end {align*}
The equation is satisfied since both sides are zero. Therefore the first solution to the ode \(z'' = r z\) is \begin {align*} z_1(x) &= p e^{ \int \omega \,dx} \\ &= {\mathrm e}^{\int \left (-\frac {i}{x^{3}}+\frac {3}{2 x}\right )d x}\\ &= x^{\frac {3}{2}} {\mathrm e}^{\frac {i}{2 x^{2}}} \end {align*}
The first solution to the original ode in \(y\) is found from \begin{align*} y_1 &= z_1 e^{ \int -\frac {1}{2} \frac {B}{A} \,dx} \\ &= z_1 e^{ -\int \frac {1}{2} \frac {3 x^{5}}{x^{6}} \,dx} \\ &= z_1 e^{-\frac {3 \ln \left (x \right )}{2}} \\ &= z_1 \left (\frac {1}{x^{\frac {3}{2}}}\right ) \\ \end{align*} Which simplifies to \[ y_1 = {\mathrm e}^{\frac {i}{2 x^{2}}} \] The second solution \(y_2\) to the original ode is found using reduction of order \[ y_2 = y_1 \int \frac { e^{\int -\frac {B}{A} \,dx}}{y_1^2} \,dx \] Substituting gives \begin{align*} y_2 &= y_1 \int \frac { e^{\int -\frac {3 x^{5}}{x^{6}} \,dx}}{\left (y_1\right )^2} \,dx \\ &= y_1 \int \frac { e^{-3 \ln \left (x \right )}}{\left (y_1\right )^2} \,dx \\ &= y_1 \left (-\frac {i {\mathrm e}^{-\frac {i}{x^{2}}}}{2}\right ) \\ \end{align*} Therefore the solution is
\begin{align*} y &= c_{1} y_1 + c_{2} y_2 \\ &= c_{1} \left ({\mathrm e}^{\frac {i}{2 x^{2}}}\right ) + c_{2} \left ({\mathrm e}^{\frac {i}{2 x^{2}}}\left (-\frac {i {\mathrm e}^{-\frac {i}{x^{2}}}}{2}\right )\right ) \\ \end{align*} This is second order nonhomogeneous ODE. Let the solution be \[ y = y_h + y_p \] Where \(y_h\) is the solution to the homogeneous ODE \( A y''(x) + B y'(x) + C y(x) = 0\), and \(y_p\) is a particular solution to the nonhomogeneous ODE \(A y''(x) + B y'(x) + C y(x) = f(x)\). \(y_h\) is the solution to \[ x^{6} y^{\prime \prime }+3 x^{5} y^{\prime }+y = 0 \] The homogeneous solution is found using the Kovacic algorithm which results in \[ y_h = c_{1} {\mathrm e}^{\frac {i}{2 x^{2}}}-\frac {i c_{2} {\mathrm e}^{-\frac {i}{2 x^{2}}}}{2} \] 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 &= {\mathrm e}^{\frac {i}{2 x^{2}}} \\ y_2 &= -\frac {i {\mathrm e}^{-\frac {i}{2 x^{2}}}}{2} \\ \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 \[ W = \begin {vmatrix} {\mathrm e}^{\frac {i}{2 x^{2}}} & -\frac {i {\mathrm e}^{-\frac {i}{2 x^{2}}}}{2} \\ \frac {d}{dx}\left ({\mathrm e}^{\frac {i}{2 x^{2}}}\right ) & \frac {d}{dx}\left (-\frac {i {\mathrm e}^{-\frac {i}{2 x^{2}}}}{2}\right ) \end {vmatrix} \] Which gives \[ W = \begin {vmatrix} {\mathrm e}^{\frac {i}{2 x^{2}}} & -\frac {i {\mathrm e}^{-\frac {i}{2 x^{2}}}}{2} \\ -\frac {i {\mathrm e}^{\frac {i}{2 x^{2}}}}{x^{3}} & \frac {{\mathrm e}^{-\frac {i}{2 x^{2}}}}{2 x^{3}} \end {vmatrix} \] Therefore \[ W = \left ({\mathrm e}^{\frac {i}{2 x^{2}}}\right )\left (\frac {{\mathrm e}^{-\frac {i}{2 x^{2}}}}{2 x^{3}}\right ) - \left (-\frac {i {\mathrm e}^{-\frac {i}{2 x^{2}}}}{2}\right )\left (-\frac {i {\mathrm e}^{\frac {i}{2 x^{2}}}}{x^{3}}\right ) \] Which simplifies to \[ W = \frac {{\mathrm e}^{\frac {i}{2 x^{2}}} {\mathrm e}^{-\frac {i}{2 x^{2}}}}{x^{3}} \] Which simplifies to \[ W = \frac {1}{x^{3}} \] Therefore Eq. (2) becomes \[ u_1 = -\int \frac {-\frac {i {\mathrm e}^{-\frac {i}{2 x^{2}}}}{2 x^{2}}}{x^{3}}\,dx \] Which simplifies to \[ u_1 = - \int -\frac {i {\mathrm e}^{-\frac {i}{2 x^{2}}}}{2 x^{5}}d x \] Hence \[ u_1 = \textit {undefined} \] And Eq. (3) becomes \[ u_2 = \int \frac {\frac {{\mathrm e}^{\frac {i}{2 x^{2}}}}{x^{2}}}{x^{3}}\,dx \] Which simplifies to \[ u_2 = \int \frac {{\mathrm e}^{\frac {i}{2 x^{2}}}}{x^{5}}d x \] Hence \[ u_2 = \textit {undefined} \] Therefore the particular solution, from equation (1) is \[ y_p(x) = \operatorname {undefined} \,{\mathrm e}^{\frac {i}{2 x^{2}}}-i \operatorname {undefined} \,{\mathrm e}^{-\frac {i}{2 x^{2}}} \] Which simplifies to \[ y_p(x) = \left (i {\mathrm e}^{-\frac {i}{2 x^{2}}}+{\mathrm e}^{\frac {i}{2 x^{2}}}\right ) \operatorname {undefined} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (c_{1} {\mathrm e}^{\frac {i}{2 x^{2}}}-\frac {i c_{2} {\mathrm e}^{-\frac {i}{2 x^{2}}}}{2}\right ) + \left (\left (i {\mathrm e}^{-\frac {i}{2 x^{2}}}+{\mathrm e}^{\frac {i}{2 x^{2}}}\right ) \operatorname {undefined}\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} {\mathrm e}^{\frac {i}{2 x^{2}}}-\frac {i c_{2} {\mathrm e}^{-\frac {i}{2 x^{2}}}}{2}+\left (i {\mathrm e}^{-\frac {i}{2 x^{2}}}+{\mathrm e}^{\frac {i}{2 x^{2}}}\right ) \operatorname {undefined} \\ \end{align*}
Verification of solutions
\[ y = c_{1} {\mathrm e}^{\frac {i}{2 x^{2}}}-\frac {i c_{2} {\mathrm e}^{-\frac {i}{2 x^{2}}}}{2}+\left (i {\mathrm e}^{-\frac {i}{2 x^{2}}}+{\mathrm e}^{\frac {i}{2 x^{2}}}\right ) \operatorname {undefined} \] 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 checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] <- linear_1 successful <- solving first the homogeneous part of the ODE successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 24
dsolve(x^6*diff(y(x),x$2)+3*x^5*diff(y(x),x)+y(x)=1/x^2,y(x), singsol=all)
\[ y \left (x \right ) = \sin \left (\frac {1}{2 x^{2}}\right ) c_{2} +\cos \left (\frac {1}{2 x^{2}}\right ) c_{1} +\frac {1}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.099 (sec). Leaf size: 32
DSolve[x^6*y''[x]+3*x^5*y'[x]+y[x]==1/x^2,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{x^2}+c_1 \cos \left (\frac {1}{2 x^2}\right )-c_2 \sin \left (\frac {1}{2 x^2}\right ) \]