Internal problem ID [5585]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz.
McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 2. Second-Order Linear Equations. Section 2.4. THE USE OF A KNOWN
SOLUTION TO FIND ANOTHER. Page 74
Problem number: 1(b).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-y=0} \end {gather*} Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{x} \end {align*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 15
dsolve([diff(y(x),x$2)-y(x)=0,exp(x)],y(x), singsol=all)
\[ y \relax (x ) = {\mathrm e}^{-x} c_{1}+c_{2} {\mathrm e}^{x} \]
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 20
DSolve[y''[x]-y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 e^x+c_2 e^{-x} \\ \end{align*}