91Ó°ÊÓ

Find a particular solution \(y_{p}\) of the given equation. In all these problems, primes denote derivatives with respect to \(x\). $$ y^{(3)}+y^{\prime}=2-\sin x $$

Find a particular solution \(y_{p}\) of the given equation. In all these problems, primes denote derivatives with respect to \(x\). $$ y^{\prime \prime}+2 y^{\prime}+5 y=e^{x} \sin x $$

In Problems, a thind-order homogeneous linear equation and three linearly independent solutions are given. Find a particular solution satisfying the given initial conditions. $$ \begin{aligned} &y^{(3)}+2 y^{\sigma}-y^{\prime}-2 y=0 ; y(0)=1, y^{\prime}(0)=2, y^{\prime \prime}(0)=0 \\ &y_{1}=e^{x}, y_{2}=e^{-x}, y_{3}=e^{-2 r} \end{aligned} $$

In Problems, a thind-order homogeneous linear equation and three linearly independent solutions are given. Find a particular solution satisfying the given initial conditions. $$ \begin{aligned} &y^{(3)}-3 y^{\prime \prime}+3 y^{\prime}-y=0 ; y(0)=2, y^{\prime}(0)=0, y^{\prime \prime}(0)=0 \\ &y_{1}=e^{x}, y_{2}=x e^{x}, y_{3}=x^{2} e^{x} \end{aligned} $$

Find a particular solution \(y_{p}\) of the given equation. In all these problems, primes denote derivatives with respect to \(x\). $$ y^{(4)}-5 y^{\prime \prime}+4 y=e^{x}-x e^{2 x} $$

According to Eq. (21), the amplitude of forced steady periodic oscillations for the system \(m x^{\prime \prime}+c x^{\prime}+k x=\) \(F_{0} \cos \omega t\) is given by $$ C(\omega)=\frac{F_{0}}{\sqrt{\left(k-m \omega^{2}\right)^{2}+(c \omega)^{2}}} . $$ (a) If \(c \geqq c_{\mathrm{cr}} / \sqrt{2}\), where \(c_{\mathrm{cr}}=\sqrt{4 \mathrm{~km}}\), show that \(C\) steadily decreases as \(\omega\) increases. \(\quad\) (b) If \(c

Problems pertain to the solution of differential equations with complex coefficients. (a) Use Euler's formula to show that every complex number can be written in the form \(r e^{i \theta}\), where \(r \geqq 0\) and \(-\pi<\theta \leqq \pi\) (b) Express the numbers \(4,-2,3 i\) \(1+i\), and \(-1+i \sqrt{3}\) in the form \(r e^{i \theta}\). (c) The two square roots of \(r e^{i \theta}\) are \(\pm \sqrt{r} e^{i \theta / 2} .\) Find the square roots of the numbers \(2-2 i \sqrt{3}\) and \(-2+2 i \sqrt{3}\).

Each of Problems 43 through 48 gives a general solution \(y(x)\) of a homogeneous second-order differential equation \(a y^{\prime \prime}+b y^{\prime}+c y=0\) with constant coefficients. Find such an equation. $$ y(x)=c_{1}+c_{2} x $$