13.10.7 problem 7

Internal problem ID [2408]
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
Date solved : Monday, January 27, 2025 at 05:51:55 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-3 y^{\prime }+2 y&=\sqrt {1+t} \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.106 (sec). Leaf size: 90

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)
 

\[ y = -\frac {\sqrt {2}\, {\mathrm e}^{2 t +2} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {2}\right )}{8}+\frac {{\mathrm e}^{2 t}}{2}+\frac {\sqrt {\pi }\, \operatorname {erf}\left (\sqrt {2}\, \sqrt {t +1}\right ) \sqrt {2}\, {\mathrm e}^{2 t +2}}{8}+\frac {\sqrt {t +1}}{2}-\frac {\sqrt {\pi }\, \operatorname {erf}\left (\sqrt {t +1}\right ) {\mathrm e}^{t +1}}{2}+\frac {\operatorname {erf}\left (1\right ) {\mathrm e}^{t +1} \sqrt {\pi }}{2}-{\mathrm e}^{t} \]

✓ Solution by Mathematica

Time used: 0.432 (sec). Leaf size: 116

DSolve[{D[y[t],{t,2}]-3*D[y[t],t]+2*y[t]==Sqrt[1+t],{y[0]==0,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{8} \left (-4 \sqrt {\pi } e^{t+1} \text {erf}\left (\sqrt {t+1}\right )+\sqrt {2 \pi } e^{2 t+2} \text {erf}\left (\sqrt {2} \sqrt {t+1}\right )-\sqrt {2 \pi } \text {erf}\left (\sqrt {2}\right ) e^{2 t+2}+4 \sqrt {\pi } \text {erf}(1) e^{t+1}-8 e^t+4 e^{2 t}+4 \sqrt {t+1}\right ) \]