11.8. Differential Equations Lab#
11.8.1. Section 2 Questions#
List the type, order, and parameters of each of the following differential equations. For each ordinary differential equation, determine if it is linear or nonlinear.
\(\dfrac{dy}{dx} +3x^2 = sin(x) + 3\)
\((\dfrac{dy}{dx})^2 +3x^2 = sin(x) + 3\)
\(\dfrac{dy}{dx}k +3x^2 k = ysin(x) + 3\)
\(\dfrac{d^2y}{dx^2} +3x^2 = sin(x) + 3\)
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List the type, order, and parameters of each of the following differential equations. For each ordinary differential equation, determine if it is linear or nonlinear.
\(\frac{\partial w}{\partial u} + \frac{\partial^2 w}{\partial v^2} = u\)
\(\frac{\partial w}{\partial u} + \frac{\partial w}{\partial v} = u^2\)
\(\dfrac{dt}{dv}t = tan(t) +9\)
\(\dfrac{dt}{dv}v = tan(t) +9\)
\( \)
Show that \(y = e^{2t}\) is a solution for all \(t\) to \(y''+y = 5y\).
Show that \(y = te^{5t}\) is a solution for all \(t\) to \(y''-10y'=-25y\)
Show that \(y = sin(3t)+cos(3t) + t^3 -6\) is a solution for all \(t\) to \(y'''+9y' = 27t^2+6\)
Show that \(y = 12t^2+t^4\) is a solution for all \(t\) to \((y'+y'')-(y'''+y^{(4)}) = y-t^4+4t^3\)
11.8.2. Section 3 Questions#
Solve the separable equations:
\( \dfrac{dx}{dt} = \dfrac{1}{t}e^{-2x} \)
\( \dfrac{dx}{dt} = \dfrac{-1}{x^2t^2} \)
\( \dfrac{dy}{dx} = cos^2(y)e^{x^2}2x \)
\( \dfrac{1}{x}-\dfrac{dx}{dt}=x^{-1}\dfrac{1}{sec(t)} \)
Solve the linear equations using the method of integrating factors (\(\mu(t) = e^{P(t)}\))
\( t\dfrac{dx}{dt} = sin(t) -x \)
\( \dfrac{dx}{dt} + x 4t^3 = \dfrac{1}{t e^{t^4}} \)
\( \dfrac{dy}{dx} + \dfrac{y}{x} = \dfrac{1}{x+x^3} \)
\( \dfrac{dy}{dx}x + \dfrac{-2y}{x^2} = xe^{-x^{-2}}sec(x)tan(x) \)
Plot the direction fields
\( \dfrac{dy}{dx}= x^2-y^2-y \)
\( \dfrac{dy}{dx}= x(y-x) \)
\( \dfrac{dy}{dx}= y(x-y) \)
Find the equilibrium solutions and draw the phase lines for the following functions. Classify each equilibrium solution.
\( y'=(y-6)(y+7) \)
\( y'= y^3-8y^2+12y \)
\( y'= y(y^2-1) \)
\( y'= y^2(y^2-1) \)
11.8.3. Section 4 Questions#
Solve the initial value problem:
\( x'' - x' -42x = 0 \); \( x'(0) = 0\), \( x''(0) = 1\)
\( x''-14x'+48x=0 \); \(x'(0)=0\), \(x''(0)=5\)
\( x''+6x'+9x=0, x'(0)=0, x''(0)=9 \)
\( x''+16x'+64x=0, x'(0)=0, x''(0)=8 \)
\( x''+4x'+8x=0, x'(0)=0, x''(0)=4 \)
\( x''+10x'+30x=0, x'(0)= 0, x''(0)= 30 \)
Solve the non-homogeneous equations:
\( x''-9x'+20x=-10 \)
\( x''-9x'+20x=20t^2 \)
\( x''-9x'+20x=e^{3t} \)
\( x''-9x'+20x=e^{5t} \)
Solve the non-homogeneous equations:
\( x''+4x'+4x =sin(t)\)
\( x''+4x'+4x =8t^2 \)
11.8.4. Section 5 Questions#
Rewrite as a system of first order equations:
\( x'''+3x''+12x'+x=sin(t)+e^{3t} \)
Test the following sets of vectors for linear independence:
\(\biggl\{\begin{bmatrix} 2e^{t} \\ e^{12t} \end{bmatrix}, \begin{bmatrix} 12e^{2t} \\ 6e^{12t} \end{bmatrix}\biggr\}\)
\(\biggl\{\begin{bmatrix} 5e^{3t} \\ 6e^{6t} \end{bmatrix}, \begin{bmatrix} 10e^{3t} \\ 12e^{6t} \end{bmatrix}\biggr\}\)
\(\biggl\{\begin{bmatrix} 3t \\ 1 \end{bmatrix}, \begin{bmatrix} 3t \\ tan^2(t) \end{bmatrix}, \begin{bmatrix} 6t \\ sec^2(t) \end{bmatrix} \biggr\}\)
Find the eigenvalues and eigenvectors:
\(\biggl\{\begin{bmatrix} 3&0 \\ 0&4 \end{bmatrix}\biggr\}\)
\(\biggl\{\begin{bmatrix} 6&2 \\ -2&2 \end{bmatrix}\biggr\}\)
\(\biggl\{\begin{bmatrix} 3&2 \\ -9&-3 \end{bmatrix}\biggr\}\)
Solve the linear system of differential equations:
\(\biggl\{\begin{bmatrix} 3&-2 \\ 4&9 \end{bmatrix}\biggr\}\)
\(\biggl\{\begin{bmatrix} 5&-2 \\ 2&1 \end{bmatrix}\biggr\}\)
\(\biggl\{\begin{bmatrix} 6&13 \\ -4&-6 \end{bmatrix}\biggr\}\)
Edit the code from chapter 5.4 to plot the phase planes, then find the equilibrium solutions:
\( \dfrac{dx}{dt}= x (x + 3) - 2xy \), \( \dfrac{dy}{dt} = y (3y - 8) - xy \)
\( \dfrac{dx}{dt} = x (5 - 2x) + xy\), \( \dfrac{dy}{dt} = y (y + 1) + y\)
Find the nullclines, equilibrium solutions, and linearization of the system:
\( x'=x(1-x)-4xy \); \( y'=y(2-y)-3xy \)
11.8.5. Section 6 Questions#
Solve the following Laplace Transforms by hand, using the formula \(\mathcal{L}(f(t)) = \int_{0}^{\infty} e^{-st}f(t) \mathrm{d}t \equiv F(s).\)
\( f(t)= sin(t)\)
\( f(t)= cos(t)\)
\( f(t)=8t+e^{3t}+cos(3t) \)
Find the inverse Laplace Transforms of the following functions:
\( \mathcal{L}(x)=-\dfrac{14}{s^3}+\dfrac{1}{s^2+49} \)
\( \mathcal{L}(x)= \dfrac{s+9}{(s+2)(s+3)}\)
\( \mathcal{L}(x)= x''+12x=3, x(0)=x'(0)=0 \)
Express the following functions in terms of the unit step function, then take the Laplace Transform:
\( f(t) = \begin{cases} 0 & \text{ if } t < 2 \\ 1 & \text{ if } t>2 \end{cases} \)
\( f(t) = \begin{cases} 2 & \text{ if } 0< t < 1 \\ 3 & \text{ if } 1<t<2 \\ 4 & \text{ if } t>2 \end{cases} \)