Toronto Math Forum
MAT2442019F => MAT244Test & Quizzes => Quiz3 => Topic started by: yuhan cheng on October 11, 2019, 02:00:00 PM

12. $y^{\prime \prime}+3 y^{\prime}=0, y(0)=2, y^{\prime}(0)=3$
Find roots of characteristic equation:
$$
\begin{array}{l}{r^{2}+3 r=0} \\ {r(r+3)=0} \\ {r_{1}=0} \\ {r_{2}=3}\end{array}
$$
General solution of equation is:
$$
y=c_{1}+c_{2} e^{3 t}
$$
Initial terms:
$$
\begin{array}{l}{y(0)=2} \\ {2=c_{1}+c_{2} e^{0}} \\ {2=c_{1}+c_{2}}\end{array}
$$
$$
\begin{array}{l}{y^{\prime}(0)=3} \\ {3 c_{2} e^{3 t}=3} \\ {3 c_{2} e^{0}=3} \\ {3 c_{2}=3}\end{array}
$$
$$
\begin{array}{l}{c_{2}=1} \\ {c_{1}+c_{2}=2} \\ {c_{1}=2+1=1}\end{array}
$$
Solution of problem: $$y=1e^{3 t}$$