Internal problem ID [11334]
Internal file name [OUTPUT/10320_Tuesday_December_27_2022_04_07_35_AM_56107345/index.tex]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter IX, Miscellaneous methods for solving equations of higher order than first.
Article 61. Transformation of variables. Page 143
Problem number: Ex 4.
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.
\[ \boxed {\sin \left (x \right )^{2} y^{\prime \prime }-2 y=0} \]
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \sin \left (x \right )^{2} \left (\frac {d}{d x}y^{\prime }\right )-2 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 {2 y}{\sin \left (x \right )^{2}} \\ \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 {2 y}{\sin \left (x \right )^{2}}=0 \\ \bullet & {} & \textrm {Multiply by denominators of the ODE}\hspace {3pt} \\ {} & {} & \sin \left (x \right )^{2} \left (\frac {d}{d x}y^{\prime }\right )-2 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} \\ {} & {} & \sin \left (x \right )^{2} \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 )-2 y \left (t \right )=0 \\ \bullet & {} & \textrm {Simplify}\hspace {3pt} \\ {} & {} & \frac {\sin \left (x \right )^{2} \left (\frac {d}{d t}\frac {d}{d t}y \left (t \right )-\frac {d}{d t}y \left (t \right )\right )}{x^{2}}-2 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 {2 x^{2} y \left (t \right )}{\sin \left (x \right )^{2}}+\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 {2 x^{2} y \left (t \right )}{\sin \left (x \right )^{2}}-\frac {d}{d t}y \left (t \right )=0 \\ \bullet & {} & \textrm {Characteristic polynomial of ODE}\hspace {3pt} \\ {} & {} & r^{2}-\frac {2 x^{2}}{\sin \left (x \right )^{2}}-r =0 \\ \bullet & {} & \textrm {Factor the characteristic polynomial}\hspace {3pt} \\ {} & {} & \frac {r^{2} \sin \left (x \right )^{2}-r \sin \left (x \right )^{2}-2 x^{2}}{\sin \left (x \right )^{2}}=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}+8 x^{2}}}{2}}{\sin \left (x \right )}, \frac {\frac {\sin \left (x \right )}{2}-\frac {\sqrt {\sin \left (x \right )^{2}+8 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}+8 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}+8 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}+8 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}+8 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}+8 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}+8 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 {\sqrt {\sin \left (x \right )^{2}+8 x^{2}}\, \csc \left (x \right )}{2}}+c_{2} x^{-\frac {\sqrt {\sin \left (x \right )^{2}+8 x^{2}}\, \csc \left (x \right )}{2}}\right ) \end {array} \]
Maple trace Kovacic algorithm successful
`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 A Liouvillian solution exists Reducible group (found an exponential solution) Group is reducible, not completely reducible <- Kovacics algorithm successful Change of variables used: [x = 1/2*arccos(t)] Linear ODE actually solved: -u(t)+(t^2-t)*diff(u(t),t)+(t^3-t^2-t+1)*diff(diff(u(t),t),t) = 0 <- change of variables successful`
✓ Solution by Maple
Time used: 0.344 (sec). Leaf size: 31
dsolve(sin(x)^2*diff(y(x),x$2)-2*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = -i \cot \left (x \right ) \ln \left (\cos \left (2 x \right )+i \sin \left (2 x \right )\right ) c_{2} -2 c_{2} +c_{1} \cot \left (x \right ) \]
✓ Solution by Mathematica
Time used: 0.339 (sec). Leaf size: 46
DSolve[Sin[x]^2*y''[x]-2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {\cos (x) \left (c_1-c_2 \log \left (\sqrt {-\sin ^2(x)}-\cos (x)\right )\right )}{\sqrt {-\sin ^2(x)}}-c_2 \]