47.5.11 problem 11

Internal problem ID [7500]
Book : Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section : Chapter 2. Linear homogeneous equations. Section 2.3.4 problems. page 104
Problem number : 11
Date solved : Sunday, March 30, 2025 at 12:10:52 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} \left (\cos \left (x \right )+\sin \left (x \right )\right ) y^{\prime \prime }-2 \cos \left (x \right ) y^{\prime }+\left (\cos \left (x \right )-\sin \left (x \right )\right ) y&=\left (\cos \left (x \right )+\sin \left (x \right )\right )^{2} {\mathrm e}^{2 x} \end{align*}

✓ Maple. Time used: 0.457 (sec). Leaf size: 321

ode:=(cos(x)+sin(x))*diff(diff(y(x),x),x)-2*cos(x)*diff(y(x),x)+(cos(x)-sin(x))*y(x) = (cos(x)+sin(x))^2*exp(2*x); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Mathematica. Time used: 4.211 (sec). Leaf size: 476

ode=(Cos[x]+Sin[x])*D[y[x],{x,2}]-2*Cos[x]*D[y[x],x]+(Cos[x]-Sin[x])*y[x]==(Cos[x]+Sin[x])^2*Exp[2*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \left (e^{-2 i x}\right )^{\frac {1}{2}-\frac {i}{2}} \left (e^{i x}\right )^{1-2 i} \left (-\frac {i \left (-1+e^{2 i \arctan \left (e^{-2 i x}\right )}\right )}{1+e^{2 i \arctan \left (e^{-2 i x}\right )}}\right )^{-\frac {1}{2}-\frac {i}{2}} \left (-i \left (e^{-2 i x}\right )^i \sqrt {1+e^{-4 i x}} \sqrt {1+e^{4 i x}} e^{2 i \left (2 x+\arctan \left (e^{-2 i x}\right )\right )}-2 i \sqrt {-e^{4 i x}} \sqrt {-\left (1+e^{4 i x}\right )^2} e^{2 i \arctan \left (e^{-2 i x}\right )} \left (-\frac {i \left (-1+e^{2 i \arctan \left (e^{-2 i x}\right )}\right )}{1+e^{2 i \arctan \left (e^{-2 i x}\right )}}\right )^i+e^{4 i x} \left (e^{-2 i x}\right )^i \sqrt {1+e^{-4 i x}} \sqrt {1+e^{4 i x}}\right )}{\sqrt {-e^{4 i x}} \sqrt {-\left (1+e^{4 i x}\right )^2} \left (1+e^{2 i \arctan \left (e^{-2 i x}\right )}\right )}+\frac {c_2 e^{3 i x} \left (e^{-2 i x}\right )^{\frac {1}{2}+\frac {i}{2}} \sqrt {1+e^{-4 i x}} \left (e^{2 i \arctan \left (e^{-2 i x}\right )}+i\right ) \left (-\frac {i \left (-1+e^{2 i \arctan \left (e^{-2 i x}\right )}\right )}{1+e^{2 i \arctan \left (e^{-2 i x}\right )}}\right )^{\frac {1}{2}-\frac {i}{2}}}{\sqrt {1+e^{4 i x}} \left (-1+e^{2 i \arctan \left (e^{-2 i x}\right )}\right )}+c_1 \left (e^{i x}\right )^{-i} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-sin(x) + cos(x))*y(x) - (sin(x) + cos(x))**2*exp(2*x) + (sin(x) + cos(x))*Derivative(y(x), (x, 2)) - 2*cos(x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE y(x)*tan(x)/2 - y(x)/2 + exp(2*x)*sin(x) + exp(2*x)/(2*cos(x)) - tan(x)*Derivative(y(x), (x, 2))/2 + Derivative(y(x), x) - Derivative(y(x), (x, 2))/2 cannot be solved by the factorable group method