Internal problem ID [1760]
Book: Differential equations and their applications, 3rd ed., M. Braun
Section: Section 2.4, The method of variation of parameters. Page 154
Problem number: 7.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-3 y^{\prime }+2 y-\sqrt {t +1}=0} \end {gather*} With initial conditions \begin {align*} [y \relax (0) = 0, y^{\prime }\relax (0) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.093 (sec). Leaf size: 72
dsolve([diff(y(t),t$2)-3*diff(y(t),t)+2*y(t)=sqrt(1+t),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
\[ y \relax (t ) = \frac {\left (\sqrt {2}\, \left (\erf \left (\sqrt {2}\, \sqrt {t +1}\right )-\erf \left (\sqrt {2}\right )\right ) {\mathrm e}^{2 t +2}-4 \,{\mathrm e}^{t +1} \left (\erf \left (\sqrt {t +1}\right )-\erf \relax (1)\right )\right ) \sqrt {\pi }}{8}-{\mathrm e}^{t}+\frac {{\mathrm e}^{2 t}}{2}+\frac {\sqrt {t +1}}{2} \]
✓ Solution by Mathematica
Time used: 0.256 (sec). Leaf size: 102
DSolve[{y''[t]-3*y'[t]+2*y[t]==Sqrt[1+t],{y[0]==0,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {1}{8} \left (4 e^t \left (-e \sqrt {\pi } \text {Erf}\left (\sqrt {t+1}\right )+e \sqrt {\pi } \text {Erf}(1)-2\right )+e^{2 t} \left (e^2 \sqrt {2 \pi } \text {Erf}\left (\sqrt {2} \sqrt {t+1}\right )-e^2 \sqrt {2 \pi } \text {Erf}\left (\sqrt {2}\right )+4\right )+4 \sqrt {t+1}\right ) \\ \end{align*}