Search
- Filter Results
- Location
- Classification
- Include attachments
- 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/Bookshelves/Linear_Algebra/Map%3A_Linear_Algebra_(Waldron_Cherney_and_Denton)/12%3A_Eigenvalues_and_Eigenvectors/12.02%3A_The_Eigenvalue-Eigenvector_EquationThe left hand side of this equation is a polynomial in the variable \(\lambda\) called the \(\textit{characteristic polynomial}\) \(P_{M}(\lambda)\) of \(M\). \[P_{M}(\lambda)=(\lambda-\lambda_{1})(\l...The left hand side of this equation is a polynomial in the variable \(\lambda\) called the \(\textit{characteristic polynomial}\) \(P_{M}(\lambda)\) of \(M\). \[P_{M}(\lambda)=(\lambda-\lambda_{1})(\lambda-\lambda_{2})\cdots(\lambda-\lambda_{n})\: \Longrightarrow\: P_{M}(\lambda_{i})=0\] An obvious candidate is the exponential function, \(e^{\lambda x}\); indeed, \(\frac{d}{dx} e^{\lambda x} = \lambda e^{\lambda x}\).
- https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/A_First_Journey_Through_Linear_Algebra/08%3A_Spectral_Theory/8.05%3A_Supplemental_Notes_-_More_on_Eigenvalues_and__Intro_to_Eigenspaces/8.5.01%3A_The_Eigenvalue-Eigenvector_EquationThe left hand side of this equation is a polynomial in the variable \(\lambda\) called the \(\textit{characteristic polynomial}\) \(P_{M}(\lambda)\) of \(M\). \[P_{M}(\lambda)=(\lambda-\lambda_{1})(\l...The left hand side of this equation is a polynomial in the variable \(\lambda\) called the \(\textit{characteristic polynomial}\) \(P_{M}(\lambda)\) of \(M\). \[P_{M}(\lambda)=(\lambda-\lambda_{1})(\lambda-\lambda_{2})\cdots(\lambda-\lambda_{n})\: \Longrightarrow\: P_{M}(\lambda_{i})=0\] An obvious candidate is the exponential function, \(e^{\lambda x}\); indeed, \(\frac{d}{dx} e^{\lambda x} = \lambda e^{\lambda x}\).
- https://math.libretexts.org/Bookshelves/Analysis/Supplemental_Modules_(Analysis)/Ordinary_Differential_Equations/3%3A_Second_Order_Linear_Differential_Equations/3.4%3A_Complex_Roots_of_the_Characteristic_EquationWe have already addressed how to solve a second order linear homogeneous differential equation with constant coefficients where the roots of the characteristic equation are real and distinct. We will ...We have already addressed how to solve a second order linear homogeneous differential equation with constant coefficients where the roots of the characteristic equation are real and distinct. We will now explain how to handle these differential equations when the roots are complex.
- https://math.libretexts.org/Courses/Reedley_College/Differential_Equations_and_Linear_Algebra_(Zook)/10%3A_Linear_Second_Order_Equations/10.02%3A_Constant_Coefficient_Homogeneous_EquationsThis section deals with homogeneous equations of the special form ay″+by′+cy=0, where a, b, and c are constant (a≠0). When you've completed this section you'll know everything there is to know about s...This section deals with homogeneous equations of the special form ay″+by′+cy=0, where a, b, and c are constant (a≠0). When you've completed this section you'll know everything there is to know about solving such equations.
- https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/17%3A_Second-Order_Differential_Equations/17.01%3A_Second-Order_Linear_EquationsWe often want to find a function (or functions) that satisfies the differential equation. The technique we use to find these solutions varies, depending on the form of the differential equation with w...We often want to find a function (or functions) that satisfies the differential equation. The technique we use to find these solutions varies, depending on the form of the differential equation with which we are working. Second-order differential equations have several important characteristics that can help us determine which solution method to use. In this section, we examine some of these characteristics and the associated terminology.
- https://math.libretexts.org/Bookshelves/Differential_Equations/Elementary_Differential_Equations_with_Boundary_Value_Problems_(Trench)/05%3A_Linear_Second_Order_Equations/5.02%3A_Constant_Coefficient_Homogeneous_EquationsThis section deals with homogeneous equations of the special form ay″+by′+cy=0, where a, b, and c are constant (a≠0). When you've completed this section you'll know everything there is to know about s...This section deals with homogeneous equations of the special form ay″+by′+cy=0, where a, b, and c are constant (a≠0). When you've completed this section you'll know everything there is to know about solving such equations.
- https://math.libretexts.org/Courses/City_College_of_San_Francisco/CCSF_Calculus/17%3A_Second-Order_Differential_Equations/17.02%3A_Second-Order_Linear_EquationsWe often want to find a function (or functions) that satisfies the differential equation. The technique we use to find these solutions varies, depending on the form of the differential equation with w...We often want to find a function (or functions) that satisfies the differential equation. The technique we use to find these solutions varies, depending on the form of the differential equation with which we are working. Second-order differential equations have several important characteristics that can help us determine which solution method to use. In this section, we examine some of these characteristics and the associated terminology.
- https://math.libretexts.org/Under_Construction/Purgatory/Differential_Equations_and_Linear_Algebra_(Zook)/17%3A_Eigenvalues_and_Eigenvectors/17.02%3A_The_Eigenvalue-Eigenvector_EquationThe left hand side of this equation is a polynomial in the variable \(\lambda\) called the \(\textit{characteristic polynomial}\) \(P_{M}(\lambda)\) of \(M\). \[P_{M}(\lambda)=(\lambda-\lambda_{1})(\l...The left hand side of this equation is a polynomial in the variable \(\lambda\) called the \(\textit{characteristic polynomial}\) \(P_{M}(\lambda)\) of \(M\). \[P_{M}(\lambda)=(\lambda-\lambda_{1})(\lambda-\lambda_{2})\cdots(\lambda-\lambda_{n})\: \Longrightarrow\: P_{M}(\lambda_{i})=0\] An obvious candidate is the exponential function, \(e^{\lambda x}\); indeed, \(\frac{d}{dx} e^{\lambda x} = \lambda e^{\lambda x}\).
- https://math.libretexts.org/Courses/Chabot_College/Math_4%3A_Differential_Equations_(Dinh)/04%3A_Linear_Higher_Order_Differential__Equations/4.03%3A_Second_Order_Constant_Coefficient_Homogeneous_EquationsThis section deals with homogeneous equations of the special form ay″+by′+cy=0, where a, b, and c are constant (a≠0). When you've completed this section you'll know everything there is to know about s...This section deals with homogeneous equations of the special form ay″+by′+cy=0, where a, b, and c are constant (a≠0). When you've completed this section you'll know everything there is to know about solving such equations.
