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- 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.
- https://math.libretexts.org/Bookshelves/Differential_Equations/Elementary_Differential_Equations_with_Boundary_Value_Problems_(Trench)/05%3A_Linear_Second_Order_EquationsIn 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.
- 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_EquationsSince \({-b\over a}\) 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 \[y_1=e^{(\lambda + \omega i)x}=e^{\lambda x}e^{i\omega x}=e^{\lambda x}(\cos \omega x+i\sin \omega x)=e^{\lambda x}\cos \omega x+ie^{\lambda x}\sin \omega x)\nonumber\]
- 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_SubstitutionsSubstituting \(tx\) 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.
- 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_ProblemsThis section applies the Laplace transform to solve initial value problems for constant coefficient second order differential equations on (0,∞).
- 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…
- https://math.libretexts.org/Bookshelves/Differential_Equations/Elementary_Differential_Equations_with_Boundary_Value_Problems_(Trench)/01%3A_Introduction/1.02%3A_Basic_ConceptsA differential equation is an equation that contains one or more derivatives of an unknown function. The order of a differential equation is the order of the highest derivative that it contains. A dif...A differential equation is an equation that contains one or more derivatives of an unknown function. The order of a differential equation is the order of the highest derivative that it contains. A differential equation is an ordinary differential equation if it involves an unknown function of only one variable, or a partial differential equation if it involves partial derivatives of a function of more than one variable. This section introduces basic concepts and definitions.
- 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_EquationsThis section deals with nonlinear equations that are not separable, but can be transformed into separable equations by a procedure similar to variation of parameters.
- https://math.libretexts.org/Bookshelves/Differential_Equations/Elementary_Differential_Equations_with_Boundary_Value_Problems_(Trench)/01%3A_Introduction/1.02%3A_Basic_Concepts/1.2E%3A_Basic_Concepts_(Exercises)f. \(y=(c_1+c_2x)e^x+\sin x+x^2; \quad y''-2y'+y=-2 \cos x+x^2-4x+2\) a. \(y=x^2(1+\ln x); \quad y''= {3xy'-4y\over x^2}, \quad y(e)=2e^2, \quad y'(e)=5e\) b. \(y= {x^2\over3}+x-1; \quad y''= {x^2-xy'...f. \(y=(c_1+c_2x)e^x+\sin x+x^2; \quad y''-2y'+y=-2 \cos x+x^2-4x+2\) a. \(y=x^2(1+\ln x); \quad y''= {3xy'-4y\over x^2}, \quad y(e)=2e^2, \quad y'(e)=5e\) b. \(y= {x^2\over3}+x-1; \quad y''= {x^2-xy'+y+1\over x^2}, \quad y(1)= {1\over3}, \quad y'(1)= {5\over3}\) c. \(y=(1+x^2)^{-1/2}; \quad y''= {(x^2-1)y-x(x^2+1)y'\over (x^2+1)^2}, \quad y(0)=1, \; y'(0)=0\) d. \(y= {x^2\over 1-x}; \quad y''= {2(x+y)(xy'-y)\over x^3}, \quad y(1/2)=1/2, \quad y'(1/2)=3\)
- 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)...\[\begin{array}{rcl} u'_1y_1+u'_2y_2+&\cdots&+u'_ny_n=0 \\ u'_1y'_1+u'_2y'_2+&\cdots&+u'_ny'_n=0 \\ \phantom{u'_1y^{(n_1)}+u'_2y_2^{(n-1)}}&\vdots& \phantom{\cdots+u'_ny^{(n-1)}_n=q} \\ u'_1y_1^{(n-2)}+u'_2y^{(n-2)}_2+&\cdots&+u'_ny^{(n-2)}_n =0 \\ u'_1y^{(n-1)}_1+u'_2y^{(n-1)}_2+&\cdots&+u'_n y^{(n-1)}_n=f(x), \end{array}\nonumber \]
- 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_EquationsIn 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.
