84.37.6 problem 26.6

Internal problem ID [22348]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 26. Solutions of linear differential equations with constant coefficients by Laplace transform. Solved problems. Page 159
Problem number : 26.6
Date solved : Thursday, October 02, 2025 at 08:37:50 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }-3 y^{\prime }+4 y&=0 \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1 \\ y^{\prime }\left (0\right )&=5 \\ \end{align*}

✓ Maple. Time used: 0.046 (sec). Leaf size: 28

ode:=diff(diff(y(x),x),x)-3*diff(y(x),x)+4*y(x) = 0; 
ic:=[y(0) = 1, D(y)(0) = 5]; 
dsolve([ode,op(ic)],y(x),method='laplace');
 

\[ y = {\mathrm e}^{\frac {3 x}{2}} \left (\cos \left (\frac {\sqrt {7}\, x}{2}\right )+\sqrt {7}\, \sin \left (\frac {\sqrt {7}\, x}{2}\right )\right ) \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 42

ode=D[y[x],{x,2}]-3*D[y[x],x]+4*y[x]==0; 
ic={y[0]==1,Derivative[1][y][0] ==5}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{3 x/2} \left (\sqrt {7} \sin \left (\frac {\sqrt {7} x}{2}\right )+\cos \left (\frac {\sqrt {7} x}{2}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.120 (sec). Leaf size: 34

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*y(x) - 3*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(x), x), x, 0): 5} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (\sqrt {7} \sin {\left (\frac {\sqrt {7} x}{2} \right )} + \cos {\left (\frac {\sqrt {7} x}{2} \right )}\right ) e^{\frac {3 x}{2}} \]