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6.1E: Spring Problems I (Exercises)
https://math.libretexts.org/Bookshelves/Differential_Equations/Elementary_Differential_Equations_with_Boundary_Value_Problems_(Trench)/06%3A_Applications_of_Linear_Second_Order_Equations/6.01%3A_Spring_Problems_I/6.1E%3A_Spring_Problems_I_(Exercises)
An external force $F(t)=.25\sin8 t$ lb is applied to the weight, which is initially displaced $4$ inches above equilibrium and given a downward velocity of $1$ ft/s. Find the period of the simpl...An external force $F(t)=.25\sin8 t$ lb is applied to the weight, which is initially displaced $4$ inches above equilibrium and given a downward velocity of $1$ ft/s. Find the period of the simple harmonic motion of a $20$ gm mass suspended from the same spring. Also, find the amplitude of the oscillation and give formulas for the sine and cosine of the initial phase angle.
5: Linear Second Order Equations
https://math.libretexts.org/Bookshelves/Differential_Equations/Elementary_Differential_Equations_with_Boundary_Value_Problems_(Trench)/05%3A_Linear_Second_Order_Equations
In this Chapter, we study a particularly important class of second order equations. Because of their many applications in science and engineering, second order differential equation have historically ...In this Chapter, we study a particularly important class of second order equations. Because of their many applications in science and engineering, second order differential equation have historically been the most thoroughly studied class of differential equations. Research on the theory of second order differential equations continues to the present day.
5.3: Constant Coefficient Homogeneous Equations
https://math.libretexts.org/Courses/Cosumnes_River_College/Math_420%3A_Differential_Equations_(Breitenbach)/05%3A_Linear_Second_Order_Equations/5.03%3A_Constant_Coefficient_Homogeneous_Equations
Since ${-b\over a}$ is just a constant we will replace it with the constant $r$ and we get a solution of the form $y=e^{rx}\nonumber$ Since $r=-1$ and $r=-5$ are roots, $y_1=e^{-x}$ and \(...Since ${-b\over a}$ is just a constant we will replace it with the constant $r$ and we get a solution of the form $y=e^{rx}\nonumber$ Since $r=-1$ and $r=-5$ are roots, $y_1=e^{-x}$ and $y_2=e^{-5x}$ are solutions of Equation $\ref{eq:5.2.4}$ and the general solution of Equation $\ref{eq:5.2.4}$ is $y_1=e^{(\lambda + \omega i)x}=e^{\lambda x}e^{i\omega x}=e^{\lambda x}(\cos \omega x+i\sin \omega x)=e^{\lambda x}\cos \omega x+ie^{\lambda x}\sin \omega x)\nonumber$
2.4: Solving Differential Equations by Substitutions
https://math.libretexts.org/Courses/Cosumnes_River_College/Math_420%3A_Differential_Equations_(Breitenbach)/02%3A_First_Order_Equations/2.04%3A_Solving_Differential_Equations_by_Substitutions
Substituting $tx$ for $x$ and $ty$ for $y$ in $M(x,y)$ we get $M(tx,ty)=(tx)^2+(tx)(ty)=t^2(x^2+xy)= t^2M(x,y)\nonumber$ and $M(x,y)$ is a homogeneous function of degree 2. Substituting ...Substituting $tx$ for $x$ and $ty$ for $y$ in $M(x,y)$ we get $M(tx,ty)=(tx)^2+(tx)(ty)=t^2(x^2+xy)= t^2M(x,y)\nonumber$ and $M(x,y)$ is a homogeneous function of degree 2. Substituting $tx$ for $x$ and $ty$ for $y$ in $N(x,y)$ we get $N(tx,ty)=(ty)^2=t^2y^2= t^2N(x,y)\nonumber$ and $N(x,y)$ is a homogeneous function of degree 2.
8.3: Solution of Initial Value Problems
https://math.libretexts.org/Bookshelves/Differential_Equations/Elementary_Differential_Equations_with_Boundary_Value_Problems_(Trench)/08%3A_Laplace_Transforms/8.03%3A_Solution_of_Initial_Value_Problems
This section applies the Laplace transform to solve initial value problems for constant coefﬁcient second order differential equations on (0,∞).
2.2: Exact Equations
https://math.libretexts.org/Courses/Cosumnes_River_College/Math_420%3A_Differential_Equations_(Breitenbach)/02%3A_First_Order_Equations/2.02%3A_Exact_Equations
\( \newcommand{\place}{\bigskip\hrule\bigskip\noindent} \newcommand{\threecol}[3]{\left[ $\begin{array}{r}#1\\#2\\#3\end{array}$ \right]} \newcommand{\threecolj}[3]{\left[\begin{array}{r}#1\\[1\jot]#2\\[1...\( \newcommand{\place}{\bigskip\hrule\bigskip\noindent} \newcommand{\threecol}[3]{\left[ $\begin{array}{r}#1\\#2\\#3\end{array}$ \right]} \newcommand{\threecolj}[3]{\left[ $\begin{array}{r}#1\\[1\jot]#2\\[1\jot]#3\end{array}$ \right]} \newcommand{\lims}[2]{\,\bigg|_{#1}^{#2}} \newcommand{\twocol}[2]{\left[ $\begin{array}{l}#1\\#2\end{array}$ \right]} \newcommand{\ctwocol}[2]{\left[ $\begin{array}{c}#1\\#2\end{array}$ \right]} \newcommand{\cthreecol}[3]{\left[ $\begin{array}{c}#1\\#2\\#3\end{array}$ \right]} \newco…
2.4: Transformation of Nonlinear Equations into Separable Equations
https://math.libretexts.org/Bookshelves/Differential_Equations/Elementary_Differential_Equations_with_Boundary_Value_Problems_(Trench)/02%3A_First_Order_Equations/2.04%3A_Transformation_of_Nonlinear_Equations_into_Separable_Equations
This section deals with nonlinear equations that are not separable, but can be transformed into separable equations by a procedure similar to variation of parameters.
7.4: Variation of Parameters for Higher Order Equations
https://math.libretexts.org/Courses/Cosumnes_River_College/Math_420%3A_Differential_Equations_(Breitenbach)/07%3A_Linear_Higher_Order_Differential_Equations/7.04%3A_Variation_of_Parameters_for_Higher_Order_Equations
\[\begin{array}{rcl} u'_1y_1+u'_2y_2+&\cdots&+u'_ny_n=0 \\ u'_1y'_1+u'_2y'_2+&\cdots&+u'_ny'_n=0 \\ \phantom{u'_1y^{(n_1)}+u'_2y_2^{(n-1)}}&\vdots& \phantom{\cdots+u'_ny^{(n-1)}_n=q} \\ u'_1y_1^{(n-2)... $\begin{array}{rcl} u'_1y_1+u'_2y_2+&\cdots&+u'_ny_n=0 \\ u'_1y'_1+u'_2y'_2+&\cdots&+u'_ny'_n=0 \\ \phantom{u'_1y^{(n_1)}+u'_2y_2^{(n-1)}}&\vdots& \phantom{\cdots+u'_ny^{(n-1)}_n=q} \\ u'_1y_1^{(n-2)}+u'_2y^{(n-2)}_2+&\cdots&+u'_ny^{(n-2)}_n =0 \\ u'_1y^{(n-1)}_1+u'_2y^{(n-1)}_2+&\cdots&+u'_n y^{(n-1)}_n=f(x), \end{array}\nonumber$
8.1: Prelude to Series Solutions of Linear Second Order Equations
https://math.libretexts.org/Courses/Cosumnes_River_College/Math_420%3A_Differential_Equations_(Breitenbach)/08%3A_Series_Solutions_of_Linear_Second_Order_Equations/8.01%3A_Prelude_to_Series_Solutions_of_Linear_Second_Order_Equations
In this Chapter, we study a class of second order differential equations that occur in many applications, but cannot be solved in closed form in terms of elementary functions. For most equations that ...In this Chapter, we study a class of second order differential equations that occur in many applications, but cannot be solved in closed form in terms of elementary functions. For most equations that occur in applications, these are polynomials of degree two or less, although the methods that we’ll develop can be extended to the case where the coefficient functions are polynomials of arbitrary degree, or even power series that converge in some circle around the origin in the complex plane.
2.4: Existence and Uniqueness of Solutions of Nonlinear Equations
https://math.libretexts.org/Courses/Red_Rocks_Community_College/MAT_2561_Differential_Equations_with_Engineering_Applications/02%3A_First_Order_Equations/2.04%3A_Existence_and_Uniqueness_of_Solutions_of_Nonlinear_Equations
Although there are methods for solving some nonlinear equations, it is impossible to find useful formulas for the solutions of most. Whether we are looking for exact solutions or numerical approximati...Although there are methods for solving some nonlinear equations, it is impossible to find useful formulas for the solutions of most. Whether we are looking for exact solutions or numerical approximations, it is useful to know conditions that imply the existence and uniqueness of solutions of initial value problems for nonlinear equations. In this section we state such a condition and illustrate it with examples.
2: First Order Equations
https://math.libretexts.org/Courses/Red_Rocks_Community_College/MAT_2561_Differential_Equations_with_Engineering_Applications/02%3A_First_Order_Equations
In this chapter, we study first-order differential equations for which there are general methods of solution.

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