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]]
\[ \boxed {y^{\prime \prime }-3 y^{\prime }+2 y=\sqrt {t +1}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.188 (sec). Leaf size: 84
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)
\[ y \left (t \right ) = -\frac {\sqrt {2}\, {\mathrm e}^{2+2 t} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {2}\right )}{8}+\frac {{\mathrm e}^{2 t}}{2}+\frac {\sqrt {2}\, \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {2}\, \sqrt {t +1}\right ) {\mathrm e}^{2+2 t}}{8}-\frac {\sqrt {\pi }\, \operatorname {erf}\left (\sqrt {t +1}\right ) {\mathrm e}^{t +1}}{2}+\frac {\sqrt {t +1}}{2}+\frac {\operatorname {erf}\left (1\right ) {\mathrm e}^{t +1} \sqrt {\pi }}{2}-{\mathrm e}^{t} \]
✓ Solution by Mathematica
Time used: 0.508 (sec). Leaf size: 116
DSolve[{y''[t]-3*y'[t]+2*y[t]==Sqrt[1+t],{y[0]==0,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to \frac {1}{8} \left (-4 \sqrt {\pi } e^{t+1} \text {erf}\left (\sqrt {t+1}\right )+\sqrt {2 \pi } e^{2 t+2} \text {erf}\left (\sqrt {2} \sqrt {t+1}\right )-\sqrt {2 \pi } \text {erf}\left (\sqrt {2}\right ) e^{2 t+2}+4 \sqrt {\pi } \text {erf}(1) e^{t+1}-8 e^t+4 e^{2 t}+4 \sqrt {t+1}\right ) \]