69.1.114 problem 162

Internal problem ID [14267]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 162
Date solved : Tuesday, January 28, 2025 at 06:24:58 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+2 h y^{\prime }+n^{2} y&=0 \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.154 (sec). Leaf size: 93

dsolve([diff(y(x),x$2)+2*h*diff(y(x),x)+n^2*y(x)=0,y(0) = a, D(y)(0) = c],y(x), singsol=all)
 

\[ y = \frac {\left (\sqrt {h^{2}-n^{2}}\, a +a h +c \right ) {\mathrm e}^{\left (-h +\sqrt {h^{2}-n^{2}}\right ) x}-{\mathrm e}^{-\left (h +\sqrt {h^{2}-n^{2}}\right ) x} \left (-\sqrt {h^{2}-n^{2}}\, a +a h +c \right )}{2 \sqrt {h^{2}-n^{2}}} \]

✓ Solution by Mathematica

Time used: 0.036 (sec). Leaf size: 123

DSolve[{D[y[x],{x,2}]+2*h*D[y[x],x]+n^2*y[x]==0,{y[0]==a,Derivative[1][y][0] ==c}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {e^{-\left (x \left (\sqrt {h^2-n^2}+h\right )\right )} \left (a h \left (e^{2 x \sqrt {h^2-n^2}}-1\right )+a \sqrt {h^2-n^2} \left (e^{2 x \sqrt {h^2-n^2}}+1\right )+c \left (e^{2 x \sqrt {h^2-n^2}}-1\right )\right )}{2 \sqrt {h^2-n^2}} \]