Chapter 3

For the given differential equation, (a) Determine the complementary solution, $y_c(t)=c_1{y}_1(t)+c_2{y}_2(t)$. (b) Use the method of variation of parameters to construct a particular solution. Then form the general solution. y^{\prime \prime}+4 y=4

(a) Find the general solution of the differential equation. (b) Impose the initial conditions to obtain the unique solution of the initial value problem. (c) Describe the behavior of the solution $y(t)$ as $t\to -\mathrm{\infty }$ and as $t\to \mathrm{\infty }$. Does $y(t)$ approach $-\mathrm{\infty },+\mathrm{\infty }$, or a finite limit? $$y^{\prime \prime }+y^{\prime }-2y=0,y(0)=3,y^{\prime }(0)=-3$$

Write each of the complex numbers in the form $\alpha +i\beta$, where $\alpha$ and $\beta$ are real numbers. (a) $2e^{i\pi /3}$ (b) $-2\sqrt{2}{e}^{-i\pi /4}$ (c) $(2-i)e^{i3\pi /2}$ (d) $\frac{1}{2\sqrt{2}}{e}^{i7\pi /6}$ (e) ${\left(\sqrt{2}{e}^{i\pi /6}\right)}^4$

For each initial value problem, determine the largest $t$-interval on which Theorem $3.1$ guarantees the existence of a unique solution. $$y^{\prime \prime }+3t^2{y}^{\prime }+2y=\sin t,y(1)=1,y^{\prime }(1)=-1$$

(a) Verify that the given functions form a fundamental set of solutions. (b) Solve the initial value problem. $y^{\prime \prime \prime }=0;y(1)=4,y^{\prime }(1)=2,y^{\prime \prime }(1)=0$ $$y_1(t)=2,y_2(t)=t-1,y_3(t)=t^2-1$$

The given function $f(t)$ is periodic with fundamental period $T$; therefore, $f(t+T)=f(t)$. Use the information given to sketch the graph of $f(t)$ over the time interval $0\le t\le 4T$. (a) $f(t)=t(2-t),0\le t<2,T=2$ (b) \(f(t)=\left\{\begin{array}{ll}1, & 0 \leq t \leq \frac{3}{2} \ 0, & \frac{3}{2}

The $t$-interval of interest is \(-\infty

For the given differential equation, $$y^{\prime \prime }-4y=4t^2$$

Consider the differential equation $y^{\prime \prime }+{\omega }_0^{2}y=F\cos \omega t$. (a) Determine the complementary solution of this differential equation. (b) Use the method of undetermined coefficients to find a particular solution in each of the cases: (i) $\omega ={\omega }_1\ne {\omega }_0$, (ii) $\omega ={\omega }_0$.

(a) Verify that the given function, $y_P(t)$, is a particular solution of the differential equation. (b) Determine the complementary solution, $y_C(t)$. (c) Form the general solution and impose the initial conditions to obtain the unique solution of the initial value problem. $$y^{\prime \prime }-2y^{\prime }-3y=-9t-3,y(0)=1,y^{\prime }(0)=3,y_p(t)=3t-1$$