Problem 1
Find the roots of the following quadratic equations: \((a) r^{2}-3 r+9=0\) (c) \(2 x^{2}+x+8=0\) (b) \(r^{2}+2 r+17=0\) (d) \(2 x^{2}-x+1=0\)
Problem 1
Find the particular integral of each equation: (a) \(y^{\prime \prime}(t)-2 y^{\prime}(t)+5 y=2\) (b) \(y^{\prime \prime}(t)+y^{\prime}(t)=7\) \((c) y^{\prime \prime}(t)+3 y=9\) \((d) y^{\prime \prime}(t)+2 y^{\prime}(t)-y=-4\) \((e) y^{\prime \prime}(t)=12\)
Problem 1
Find the particular integral of each of the following: \((a) y^{(\prime \prime t)}(t)+2 y^{\prime \prime}(t)+y^{\prime}(t)+2 y=8\) (b) \(y^{\prime \prime \prime}(t)+y^{\prime \prime}(t)+3 y^{\prime}(t)=1\) (c) \(3 y^{\prime \prime \prime}(t)+9 y^{\prime \prime}(t)=1\) \((d) y^{(4)}(t)+y^{\prime \prime}(t)=4\)
Problem 2
Find the \(y_{p}\) and the \(y_{c}\) (and hence the general solution) of: (a) \(y^{\prime \prime \prime}(t)-2 y^{\prime \prime}(t)-y^{\prime}(t)+2 y=4\) \(\left[\text { Hint: } r^{3} \cdots 2 r^{2}-r+2=(r-1)(r+1)(r-2)\right]\) (b) \(y^{\prime \prime \prime}(t)+7 y^{\prime \prime}(t)+15 y^{\prime}(t)+9 y=0\) \(\left[\text {Hint}: r^{3}+7 r^{2}+15 r+9=(r-1)\left(r^{2}+6 r+9\right)\right]\) (c) \(y^{\prime \prime \prime \prime}(t)+6 y^{\prime \prime}(t)+10 y^{\prime}(t)-8 y=8\) \(\left[\text {Hint:} r^{3}+6 r^{2}+10 r+8=(r-4)\left(r^{2}+2 r+2\right)\right]\)
Problem 2
Find the complementary function of each equation: (a) \(y^{\prime \prime}(t)+3 y^{\prime}(t)-4 y=12\) (b) \(y^{\prime \prime}(t)+6 y^{\prime}(t)+5 y=10\) (c) \(y^{\prime \prime}(t)-2 y^{\prime}(t)+y=3\) \((d) y^{\prime \prime}(t)+8 y^{\prime}(t)+16 y=0\)
Problem 2
Find the particular integral of each of the following equations by the method of undetermined coefficients: (a) \(y^{n}(t)+2 y^{\prime}(t)+y=t\) (b) \(y^{\prime \prime}(t)+4 y^{\prime}(t)+y=2 t^{2}\) (c) \(y^{\prime \prime}(t)+y^{\prime}(t)+2 y=e^{t}\) (d) \(y^{\prime \prime}(t)+y^{\prime}(t)+3 y=\sin t\)
Problem 2
(a) How many degrees are there in a radian? (b) How many radians are there in a degree?
Problem 2
Find the \(\gamma_{p}\) and the \(y_{c \prime}\) the general solution, and the definite solution of each of the following: $$y^{\prime \prime}(t)+4 y^{\prime}(t)+8 y=2 ; y(0)=2 \frac{1}{4}, y^{\prime}(0)=4$$
Problem 3
Find the \(\gamma_{p}\) and the \(y_{c \prime}\) the general solution, and the definite solution of each of the following: $$y^{\prime \prime}(t)+3 y^{\prime}(t)-4 y=12 ; y(0)=2, y^{\prime}(0)=2$$
Problem 4
Find the \(\gamma_{p}\) and the \(y_{c \prime}\) the general solution, and the definite solution of each of the following: $$y^{\prime \prime}(t)-2 y^{\prime}(t)-10 y=5 ; y(0)=6, y^{\prime}(0)=8 \frac{1}{2}$$
