41.5.11 problem 11

Internal problem ID [8786]
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 : Tuesday, September 30, 2025 at 05:50:58 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: 0.973 (sec). Leaf size: 409

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]
 

\begin{align*} y(x)&\to \exp \left (\int _1^{e^{i x}}\left (\frac {1-\frac {i}{2}}{K[1]}-\frac {K[1]}{K[1]^2+i}\right )dK[1]-\frac {1}{2} \int _1^{e^{i x}}\left (\frac {2+i}{K[2]}-\frac {2 K[2]}{K[2]^2+i}\right )dK[2]\right ) \left (\int _1^{e^{i x}}\left (\frac {1}{2}+\frac {i}{2}\right ) \exp \left (\int _1^{K[4]}\left (\frac {1-\frac {i}{2}}{K[1]}-\frac {K[1]}{K[1]^2+i}\right )dK[1]+\frac {1}{2} \int _1^{K[4]}\left (\frac {2+i}{K[2]}-\frac {2 K[2]}{K[2]^2+i}\right )dK[2]\right ) K[4]^{-3-2 i} \left (1-i K[4]^2\right ) \int _1^{K[4]}\exp \left (-2 \int _1^{K[3]}\frac {(1+2 i)-i K[1]^2}{2 K[1]^3+2 i K[1]}dK[1]\right )dK[3]dK[4]+\int _1^{e^{i x}}\exp \left (-2 \int _1^{K[3]}\frac {(1+2 i)-i K[1]^2}{2 K[1]^3+2 i K[1]}dK[1]\right )dK[3] \left (\int _1^{e^{i x}}\left (-\frac {1}{2}+\frac {i}{2}\right ) \exp \left (\int _1^{K[5]}\left (\frac {1-\frac {i}{2}}{K[1]}-\frac {K[1]}{K[1]^2+i}\right )dK[1]+\frac {1}{2} \int _1^{K[5]}\left (\frac {2+i}{K[2]}-\frac {2 K[2]}{K[2]^2+i}\right )dK[2]\right ) K[5]^{-3-2 i} \left (K[5]^2+i\right )dK[5]+c_2\right )+c_1\right ) \end{align*}

✗ 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