Differential Equations and Linear Algebra 2nd Edition solutions

Differential Equations and Linear Algebra 2
Separable or Not Determine whether each equation in Problems \(1-10\) is separable or not. Write separable ones in separable form. Determine constant solutions, if any. \(y^{\prime}=\sin (t+y)\) Equation Transcrip

Differential Equations and Linear Algebra 2
Derivatives of Complex Functions Find the derivatives F(t) and F(t) for each of the following complex-valued functions of the real variable t. (a) \(F(t)=e^{(1-i) t}\) (b) \(F(t)=e^{3 i t}\) (c) \(F(t)=e^{(2+3 i) t}\) ________________ Equation Transcription: Text Transcriptio

Differential Equations and Linear Algebra 2
Complex Exponential Functions Write each of the following complex-valued functions in a + bi form. (a) \(e^{4 \pi i t}\) (b) \(e^{(-1+2 i) t}\) ________________ Equation Transcription: Text Transcription: e^4 piit e^(-1+2i)t

Differential Equations and Linear Algebra 2
Complex Operations Write the following complex numbers in the form \(a+bi\). (a) \((2+3 i)(4-i)\) (b) \((2+3 i)(1+i)\) (c) \(\frac{1}{1+i}\) (d) \(\frac{2+i}{3+i}\)

Differential Equations and Linear Algebra 2
Complex Verification I Verify that the two complex numbers \(z=-1 \pm i\) satisfy the equation \(z^{2}+2 z+2=0\). ________________ Equation Transcription: Text Transcription: z=-1+- i z^2+2z+2=0

Differential Equations and Linear Algebra 2
Graphing by Calculator For the combinations of sine and cosine functions in Problems 9-13, do the following. (a) Use a graphing calculator or computer to sketch the graph of each function. (b) From your graphs, estimate the amplitude. period, and phase shift \(\delta / \omega_{0}\) of the resulti

Differential Equations and Linear Algebra 2
Real and Complex Parts of Exponentials Write each of the following complex numbers in a + bi form. (a) \(e^{(1+\pi i)}\) (b) \(e^{(2+\pi i / 2)}\) (c) \(e^{\pi i}\) (d) \(e^{-\pi i}\) ________________ Equation Transcription: Text Transcription: e(1+pi i) e(2+pi i/2)

Differential Equations and Linear Algebra 2
Complex Exponential Numbers Write each of the following complex exponentials in the form a + bi. (a) \(e^{2 \pi 1}\) (b) \(e^{1 \pi / 2}\) (c) \(e^{-1 \pi}\) (d) \(e^{(2+\pi i / 4)}\) ________________ Equation Transcription: Text Transcription: e^2 pi 1 e^1pi/2 e^-1p

Differential Equations and Linear Algebra 2
Absolute Value Revisited Use the formula \(|z|=\sqrt{z \bar{z}}\) to find the absolute value \(|4+2 i|\). ________________ Equation Transcription: Text Transcription: |z|=sqrt z bar z |4+2i|

Differential Equations and Linear Algebra 2
Root of Unity Find the roots of the following equations. (a) \(z^{2}=1\) (b) \(z^{3}=1\) (c) \(z^{4}=1\) ________________ Equation Transcription: Text Transcription: z^2=1 z^3=1 z^4=1

Differential Equations and Linear Algebra 2
Magnitudes and Angles Find the absolute value and polar angle of each of the following complex numbers. (a) \(1+2 i\) (b) \(-i\) (c) \(-1-i\) (d) \(-2+3 i\) (e) \(e^{2 i}\) (f) \(\frac{2+i}{1+i}\) ________________ Equation Transcription: Text Transcription: 1+2

Differential Equations and Linear Algebra 2
Complex Verification II Show that \(\frac{1+i}{\sqrt{2}}\) satisfies the equation \(z^{4}=-1\). ________________ Equation Transcription: Text Transcription: 1+i/sqrt 2 z^4=-1

Differential Equations and Linear Algebra 2
Real and Complex Parts If \(z=a+b i\), find the following quantities in terms of a and b. (a) \(R e\left(z^{2}+2 z\right)\) (b) \(\operatorname{Im}\left(z^{2}+2 z\right)\) ________________ Equation Transcription: Text Transcription: z=a+bi Re(z^2+2z) Im(z^2+2z)

Differential Equations and Linear Algebra 2
Solutions in Particular Solve the \(IVPs\) in Problems 25-34. Sketch the trajectory. \(\vec{\mathbf{x}}^{\prime}=\left[\begin{array}{ll}1&2\\ 2&1\end{array}\right]\vec{\mathbf{x}},\ \ \ \ \ \ \ \quad\vec{\mathbf{x}}(0)=\left[\begin{array}{l}1\\ 3\end{array}\right]\)

Differential Equations and Linear Algebra 2
Dissecting Transformations In each of Problems 21-40, a transformation \(T(\bar{v})=A \bar{v}, T: R^{n} \rightarrow R^{m}\), is given by a matrix A. For each transformation, find the kernel, the image, and their dimensions. Determine whether the transformation is injective or subjective. \(\left[

Differential Equations and Linear Algebra 2
Matching Problem For Problems 21-28. determine which graph of the particular solution shown in Fig. 4.3.9 matches each differential equation. \(y^{\prime \prime}+3 y^{\prime}+2 y=0\) ________________ Equation Transcription: Text Transcription: y''+3y'+2y=0

Differential Equations and Linear Algebra 2
DE Solutions Using Matrix Exponentials Using matrix exponentials. find the general solutions of the linear systems given in Problems 7-14. \(\bar{x}^{\prime}=\left[\begin{array}{cc} -1 & 0 \\ 0 & 2 \end{array}\right]\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right]+\left[\begin{array}{l} 1

Differential Equations and Linear Algebra 2
Prove the following property (stated in this section): If A is invertible, then so is \(A^{-1}\), and \(\left(A^{-1}\right)^{-1}=A\) Equation Transcription Text Transcription A^-1 (A^-1)^-1=A

Differential Equations and Linear Algebra 2
The Base for the Space Determine whether or not the set S in each of Problems 49-56 is a basis for the specified vector space V. \(\mathbf{V}=\mathbb{P}_{3} ; \quad S=\left\{t^{2}, t+3, t^{3}+4, t-1, t^{2}-5 t+1\right\}\)

Differential Equations and Linear Algebra 2
Creating New Problems (a) Find a \(3 \times 3\) matrix with a double eigenvalue \(\lambda_{1}=\lambda_{2}\) that has only one eigenvector, and a separate eigenvalue \(\lambda_{3} \neq \lambda_{1}\) with another eigenvector. (b) Find a \(3 \times 3\) matrix with a triple eigenvalue and two linearl