69.22.10 problem 715

Internal problem ID [18476]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 17. Boundary value problems. Exercises page 163
Problem number : 715
Date solved : Thursday, October 02, 2025 at 03:14:00 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+\alpha ^{2} y&=1 \end{align*}

With initial conditions

\begin{align*} y^{\prime }\left (0\right )&=\alpha \\ y^{\prime }\left (\pi \right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.082 (sec). Leaf size: 21

ode:=diff(diff(y(x),x),x)+alpha^2*y(x) = 1; 
ic:=[D(y)(0) = alpha, D(y)(Pi) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \sin \left (\alpha x \right )+\cos \left (\alpha x \right ) \cot \left (\alpha \pi \right )+\frac {1}{\alpha ^{2}} \]

✗ Mathematica

ode=D[y[x],{x,2}]+\[Alpha]^2*D[y[x],x]==1; 
ic={Derivative[1][y][0] ==\[Alpha],Derivative[1][y][Pi]==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

{}

✓ Sympy. Time used: 0.080 (sec). Leaf size: 60

from sympy import * 
x = symbols("x") 
Alpha = symbols("Alpha") 
y = Function("y") 
ode = Eq(Alpha**2*y(x) + Derivative(y(x), (x, 2)) - 1,0) 
ics = {Subs(Derivative(y(x), x), x, 0): alpha, Subs(Derivative(y(x), x), x, pi): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {i \alpha e^{2 i \pi \mathrm {A}} e^{- i \mathrm {A} x}}{\mathrm {A} e^{2 i \pi \mathrm {A}} - \mathrm {A}} + \frac {i \alpha e^{i \mathrm {A} x}}{\mathrm {A} e^{2 i \pi \mathrm {A}} - \mathrm {A}} + \frac {1}{\mathrm {A}^{2}} \]