Internal problem ID [12279]
Internal file name [OUTPUT/10932_Thursday_September_28_2023_01_09_15_AM_7130705/index.tex]
Book: APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A.
Dobrushkin. CRC Press 2015
Section: Chapter 4, Second and Higher Order Linear Differential Equations. Problems page
221
Problem number: Problem 20(f).
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.
\[ \boxed {\left (2 \sin \left (x \right )-\cos \left (x \right )\right ) y^{\prime \prime }+\left (7 \sin \left (x \right )+4 \cos \left (x \right )\right ) y^{\prime }+10 y \cos \left (x \right )=0} \]
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 symmetry of the form [xi=0, eta=F(x)] <- linear_1 successful Change of variables used: [x = arccos(t)] Linear ODE actually solved: -10*t*u(t)+(-8*t^2+7+6*(-t^2+1)^(1/2)*t)*diff(u(t),t)+(2*(-t^2+1)^(1/2)*t^2-t^3-2*(-t^2+1)^(1/2)+t)*diff(diff(u(t),t),t) = 0 <- change of variables successful`
✓ Solution by Maple
Time used: 0.234 (sec). Leaf size: 76
dsolve((2*sin(x)-cos(x))*diff(y(x),x$2)+(7*sin(x)+4*cos(x))*diff(y(x),x)+10*y(x)*cos(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = -{\mathrm e}^{-\left (\int \frac {5 \cos \left (x \right ) \cot \left (x \right )-6 \csc \left (x \right )}{-2 \sin \left (x \right )+\cos \left (x \right )}d x \right )} \left (c_{2} \left (\int \frac {\csc \left (x \right ) {\mathrm e}^{\int \frac {5 \cos \left (x \right ) \cot \left (x \right )-6 \csc \left (x \right )}{-2 \sin \left (x \right )+\cos \left (x \right )}d x}}{-2 \sin \left (x \right )+\cos \left (x \right )}d x \right )-c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 3.823 (sec). Leaf size: 112
DSolve[(2*Sin[x]-Cos[x])*y''[x]+(7*Sin[x]+4*Cos[x])*y'[x]+10*y[x]*Cos[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {e^{2 i x} \left (c_2 \int _1^{e^{i x}}\frac {e^{3 i \arctan \left (\frac {2-2 K[1]^2}{K[1]^2+1}\right )} K[1]^{-2+2 i} \left ((1+2 i) K[1]^2+(1-2 i)\right )^4}{\left (5 K[1]^4-6 K[1]^2+5\right )^{3/2}}dK[1]+c_1\right )}{\left ((1+2 i) e^{2 i x}+(1-2 i)\right )^2} \]