Internal problem ID [1173]
Internal file name [OUTPUT/1174_Sunday_June_05_2022_02_04_09_AM_30580542/index.tex]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 5 linear second order equations. Section 5.7 Variation of Parameters. Page
262
Problem number: 19.
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, _linear, _nonhomogeneous]]
Unable to solve or complete the solution.
\[ \boxed {\sin \left (x \right ) y^{\prime \prime }+\left (2 \sin \left (x \right )-\cos \left (x \right )\right ) y^{\prime }+\left (\sin \left (x \right )-\cos \left (x \right )\right ) y={\mathrm e}^{-x}} \]
Maple trace Kovacic algorithm successful
`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)] trying symmetries linear in x and y(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 -> 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)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm A Liouvillian solution exists Reducible group (found an exponential solution) <- Kovacics algorithm successful Change of variables used: [x = arccos(t)] Linear ODE actually solved: ((-t^2+1)^(1/2)-t)*u(t)+(2*t^2-2)*diff(u(t),t)+(-t^2+1)^(3/2)*diff(diff(u(t),t),t) = 0 <- change of variables successful <- solving first the homogeneous part of the ODE successful`
✓ Solution by Maple
Time used: 0.093 (sec). Leaf size: 58
dsolve(sin(x)*diff(y(x),x$2)+(2*sin(x)-cos(x))*diff(y(x),x)+(sin(x)-cos(x))*y(x)=exp(-x),y(x), singsol=all)
\[ y \left (x \right ) = -{\mathrm e}^{\arcsin \left (\cos \left (x \right )\right )} \left (\left (\int \csc \left (x \right )^{2} {\mathrm e}^{-\arcsin \left (\cos \left (x \right )\right )-x}d x \right ) \cos \left (x \right )-\cos \left (x \right ) c_{1} -\left (\int \cot \left (x \right ) \csc \left (x \right ) {\mathrm e}^{-\arcsin \left (\cos \left (x \right )\right )-x}d x \right )-c_{2} \right ) \]
✓ Solution by Mathematica
Time used: 0.809 (sec). Leaf size: 121
DSolve[Sin[x]*y''[x]+(2*Sin[x]-Cos[x])*y'[x]+(Sin[x]-Cos[x])*y[x]==Exp[-x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\sqrt {\sin ^2(x)} \exp \left (-\arccos (\cos (x))-4 \arctan \left (\frac {\sqrt {\sin ^2(x)}}{\cos (x)+1}\right )+4 \cot ^{-1}\left (\frac {\cos (x)+1}{\sqrt {\sin ^2(x)}}\right )\right )+c_2 \cos (x) \exp \left (2 \left (\cot ^{-1}\left (\frac {\cos (x)+1}{\sqrt {\sin ^2(x)}}\right )-2 \arctan \left (\frac {\sqrt {\sin ^2(x)}}{\cos (x)+1}\right )\right )\right )+c_1 e^{-2 \cot ^{-1}\left (\frac {\cos (x)+1}{\sqrt {\sin ^2(x)}}\right )} \]