Internal problem ID [9762]
Internal file name [OUTPUT/8704_Monday_June_06_2022_05_15_54_AM_54723401/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1435.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_2nd_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime \prime }+\frac {4 \sin \left (3 x \right ) y}{\sin \left (x \right )^{3}}=0} \]
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (\frac {d}{d x}y^{\prime }\right ) \sin \left (x \right )^{3}+4 \sin \left (3 x \right ) y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & \frac {d}{d x}y^{\prime } \\ \bullet & {} & \textrm {Isolate 2nd derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime }=-\frac {4 \sin \left (3 x \right ) y}{\sin \left (x \right )^{3}} \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y\hspace {3pt}\textrm {on the lhs of the ODE and the rest on the rhs of the ODE; ODE is linear}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime }+\frac {4 \sin \left (3 x \right ) y}{\sin \left (x \right )^{3}}=0 \\ \bullet & {} & \textrm {Multiply by denominators of the ODE}\hspace {3pt} \\ {} & {} & \left (\frac {d}{d x}y^{\prime }\right ) \sin \left (x \right )^{3}+4 \sin \left (3 x \right ) y=0 \\ \bullet & {} & \textrm {Make a change of variables}\hspace {3pt} \\ {} & {} & t =\ln \left (x \right ) \\ \square & {} & \textrm {Substitute the change of variables back into the ODE}\hspace {3pt} \\ {} & \circ & \textrm {Calculate the}\hspace {3pt} \hspace {3pt}\textrm {1st}\hspace {3pt} \hspace {3pt}\textrm {derivative of}\hspace {3pt} \hspace {3pt}\textrm {y}\hspace {3pt} \hspace {3pt}\textrm {with respect to}\hspace {3pt} \hspace {3pt}\textrm {x}\hspace {3pt} \hspace {3pt}\textrm {, using the chain rule}\hspace {3pt} \\ {} & {} & y^{\prime }=\left (\frac {d}{d t}y \left (t \right )\right ) t^{\prime }\left (x \right ) \\ {} & \circ & \textrm {Compute derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\frac {d}{d t}y \left (t \right )}{x} \\ {} & \circ & \textrm {Calculate the}\hspace {3pt} \hspace {3pt}\textrm {2nd}\hspace {3pt} \hspace {3pt}\textrm {derivative of}\hspace {3pt} \hspace {3pt}\textrm {y}\hspace {3pt} \hspace {3pt}\textrm {with respect to}\hspace {3pt} \hspace {3pt}\textrm {x}\hspace {3pt} \hspace {3pt}\textrm {, using the chain rule}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime }=\left (\frac {d}{d t}\frac {d}{d t}y \left (t \right )\right ) {t^{\prime }\left (x \right )}^{2}+\left (\frac {d}{d x}t^{\prime }\left (x \right )\right ) \left (\frac {d}{d t}y \left (t \right )\right ) \\ {} & \circ & \textrm {Compute derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime }=\frac {\frac {d}{d t}\frac {d}{d t}y \left (t \right )}{x^{2}}-\frac {\frac {d}{d t}y \left (t \right )}{x^{2}} \\ & {} & \textrm {Substitute the change of variables back into the ODE}\hspace {3pt} \\ {} & {} & \left (\frac {\frac {d}{d t}\frac {d}{d t}y \left (t \right )}{x^{2}}-\frac {\frac {d}{d t}y \left (t \right )}{x^{2}}\right ) \sin \left (x \right )^{3}+4 \sin \left (3 x \right ) y \left (t \right )=0 \\ \bullet & {} & \textrm {Simplify}\hspace {3pt} \\ {} & {} & \frac {\left (\frac {d}{d t}\frac {d}{d t}y \left (t \right )-\frac {d}{d t}y \left (t \right )\right ) \sin \left (x \right )^{3}}{x^{2}}+4 \sin \left (3 x \right ) y \left (t \right )=0 \\ \bullet & {} & \textrm {Isolate 2nd derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d t}\frac {d}{d t}y \left (t \right )=-\frac {4 \sin \left (3 x \right ) x^{2} y \left (t \right )}{\sin \left (x \right )^{3}}+\frac {d}{d t}y \left (t \right ) \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y \left (t \right )\hspace {3pt}\textrm {on the lhs of the ODE and the rest on the rhs of the ODE; ODE is linear}\hspace {3pt} \\ {} & {} & \frac {d}{d t}\frac {d}{d t}y \left (t \right )+\frac {4 \sin \left (3 x \right ) x^{2} y \left (t \right )}{\sin \left (x \right )^{3}}-\frac {d}{d t}y \left (t \right )=0 \\ \bullet & {} & \textrm {Characteristic polynomial of ODE}\hspace {3pt} \\ {} & {} & r^{2}+\frac {4 \sin \left (3 x \right ) x^{2}}{\sin \left (x \right )^{3}}-r =0 \\ \bullet & {} & \textrm {Factor the characteristic polynomial}\hspace {3pt} \\ {} & {} & \frac {r^{2} \sin \left (x \right )^{3}-r \sin \left (x \right )^{3}+4 \sin \left (3 x \right ) x^{2}}{\sin \left (x \right )^{3}}=0 \\ \bullet & {} & \textrm {Roots of the characteristic polynomial}\hspace {3pt} \\ {} & {} & r =\left (\frac {\frac {\sin \left (x \right )}{2}+\frac {\sqrt {\sin \left (x \right )^{2}+64 x^{2} \sin \left (x \right )^{2}-48 x^{2}}}{2}}{\sin \left (x \right )}, \frac {\frac {\sin \left (x \right )}{2}-\frac {\sqrt {\sin \left (x \right )^{2}+64 x^{2} \sin \left (x \right )^{2}-48 x^{2}}}{2}}{\sin \left (x \right )}\right ) \\ \bullet & {} & \textrm {1st solution of the ODE}\hspace {3pt} \\ {} & {} & y_{1}\left (t \right )={\mathrm e}^{\frac {\left (\frac {\sin \left (x \right )}{2}+\frac {\sqrt {\sin \left (x \right )^{2}+64 x^{2} \sin \left (x \right )^{2}-48 x^{2}}}{2}\right ) t}{\sin \left (x \right )}} \\ \bullet & {} & \textrm {2nd solution of the ODE}\hspace {3pt} \\ {} & {} & y_{2}\left (t \right )={\mathrm e}^{\frac {\left (\frac {\sin \left (x \right )}{2}-\frac {\sqrt {\sin \left (x \right )^{2}+64 x^{2} \sin \left (x \right )^{2}-48 x^{2}}}{2}\right ) t}{\sin \left (x \right )}} \\ \bullet & {} & \textrm {General solution of the ODE}\hspace {3pt} \\ {} & {} & y \left (t \right )=c_{1} y_{1}\left (t \right )+c_{2} y_{2}\left (t \right ) \\ \bullet & {} & \textrm {Substitute in solutions}\hspace {3pt} \\ {} & {} & y \left (t \right )=c_{1} {\mathrm e}^{\frac {\left (\frac {\sin \left (x \right )}{2}+\frac {\sqrt {\sin \left (x \right )^{2}+64 x^{2} \sin \left (x \right )^{2}-48 x^{2}}}{2}\right ) t}{\sin \left (x \right )}}+c_{2} {\mathrm e}^{\frac {\left (\frac {\sin \left (x \right )}{2}-\frac {\sqrt {\sin \left (x \right )^{2}+64 x^{2} \sin \left (x \right )^{2}-48 x^{2}}}{2}\right ) t}{\sin \left (x \right )}} \\ \bullet & {} & \textrm {Change variables back using}\hspace {3pt} t =\ln \left (x \right ) \\ {} & {} & y=c_{1} {\mathrm e}^{\frac {\left (\frac {\sin \left (x \right )}{2}+\frac {\sqrt {\sin \left (x \right )^{2}+64 x^{2} \sin \left (x \right )^{2}-48 x^{2}}}{2}\right ) \ln \left (x \right )}{\sin \left (x \right )}}+c_{2} {\mathrm e}^{\frac {\left (\frac {\sin \left (x \right )}{2}-\frac {\sqrt {\sin \left (x \right )^{2}+64 x^{2} \sin \left (x \right )^{2}-48 x^{2}}}{2}\right ) \ln \left (x \right )}{\sin \left (x \right )}} \\ \bullet & {} & \textrm {Simplify}\hspace {3pt} \\ {} & {} & y=\sqrt {x}\, \left (c_{1} x^{\frac {\csc \left (x \right ) \sqrt {\sin \left (x \right )^{2}+64 x^{2} \sin \left (x \right )^{2}-48 x^{2}}}{2}}+c_{2} x^{-\frac {\csc \left (x \right ) \sqrt {\sin \left (x \right )^{2}+64 x^{2} \sin \left (x \right )^{2}-48 x^{2}}}{2}}\right ) \end {array} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) -> Trying changes of variables to rationalize or make the ODE simpler trying a quadrature 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)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm <- No Liouvillian solutions exists -> Trying a solution in terms of special functions: -> Bessel -> elliptic -> Legendre <- Legendre successful <- special function solution successful Change of variables used: [x = arccos(t)] Linear ODE actually solved: 16*(-t^2+1)^(1/2)*(4*t^2-1)*u(t)-4*(-t^2+1)^(3/2)*t*diff(u(t),t)+4*(-t^2+1)^(1/2)*(t^4-2*t^2+1)*diff(diff(u(t),t),t) = 0 <- change of variables successful`
✓ Solution by Maple
Time used: 0.234 (sec). Leaf size: 38
dsolve(diff(diff(y(x),x),x) = -4*sin(3*x)/sin(x)^3*y(x),y(x), singsol=all)
\[ y \left (x \right ) = \sqrt {\sin \left (x \right )}\, \left (c_{1} \operatorname {LegendreP}\left (-\frac {1}{2}+4 i, \frac {i \sqrt {47}}{2}, \cos \left (x \right )\right )+c_{2} \operatorname {LegendreQ}\left (-\frac {1}{2}+4 i, \frac {i \sqrt {47}}{2}, \cos \left (x \right )\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.213 (sec). Leaf size: 61
DSolve[y''[x] == -4*Csc[x]^3*Sin[3*x]*y[x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \sqrt [4]{-\sin ^2(x)} \left (c_1 P_{-\frac {1}{2}+4 i}^{\frac {i \sqrt {47}}{2}}(\cos (x))+c_2 Q_{-\frac {1}{2}+4 i}^{\frac {i \sqrt {47}}{2}}(\cos (x))\right ) \]