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- 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/Monroe_Community_College/MTH_225_Differential_Equations/05%3A_Linear_Second_Order_Equations/5.07%3A_Variation_of_ParametersThis section deals with the method traditionally called variation of parameters, which enables us to find the general solution of a nonhomogeneous linear second order equation provided that we know tw...This section deals with the method traditionally called variation of parameters, which enables us to find the general solution of a nonhomogeneous linear second order equation provided that we know two nontrivial solutions (with nonconstant ratio) of the associated homogeneous equation.
- https://math.libretexts.org/Courses/Mission_College/Math_4B%3A_Differential_Equations_(Kravets)/04%3A_Linear_Second_Order_Equations/4.10%3A_Variation_of_ParametersThis section deals with the method traditionally called variation of parameters, which enables us to find the general solution of a nonhomogeneous linear second order equation provided that we know tw...This section deals with the method traditionally called variation of parameters, which enables us to find the general solution of a nonhomogeneous linear second order equation provided that we know two nontrivial solutions (with nonconstant ratio) of the associated homogeneous equation.
- https://math.libretexts.org/Under_Construction/Purgatory/Differential_Equations_and_Linear_Algebra_(Zook)/19%3A_Linear_Higher_Order_Differential_Equations/19.04%3A_Variation_of_Parameters_for_Higher_Order_EquationsThis section extends the method of variation of parameters to higher order equations. We’ll show how to use the method of variation of parameters to find a particular solution of Ly=F, provided that ...This section extends the method of variation of parameters to higher order equations. We’ll show how to use the method of variation of parameters to find a particular solution of Ly=F, provided that we know a fundamental set of solutions of the homogeous equation: Ly=0.
- https://math.libretexts.org/Courses/Mission_College/Math_4B%3A_Differential_Equations_(Kravets)/04%3A_Linear_Second_Order_Equations/4.11%3A_Variation_of_Parameters_for_Higher_Order_EquationsThis section extends the method of variation of parameters to higher order equations. We’ll show how to use the method of variation of parameters to find a particular solution of Ly=F, provided that ...This section extends the method of variation of parameters to higher order equations. We’ll show how to use the method of variation of parameters to find a particular solution of Ly=F, provided that we know a fundamental set of solutions of the homogeous equation: Ly=0.
- https://math.libretexts.org/Bookshelves/Differential_Equations/Elementary_Differential_Equations_with_Boundary_Value_Problems_(Trench)/09%3A_Linear_Higher_Order_Differential_Equations/9.04%3A_Variation_of_Parameters_for_Higher_Order_EquationsThis section extends the method of variation of parameters to higher order equations. We’ll show how to use the method of variation of parameters to find a particular solution of Ly=F, provided that ...This section extends the method of variation of parameters to higher order equations. We’ll show how to use the method of variation of parameters to find a particular solution of Ly=F, provided that we know a fundamental set of solutions of the homogeous equation: Ly=0.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/03%3A_Higher_order_linear_ODEsIn this chapter, we go a little further and look at second-order equations, which are equations containing second derivatives of the dependent variable. The solution methods we examine are different f...In this chapter, we go a little further and look at second-order equations, which are equations containing second derivatives of the dependent variable. The solution methods we examine are different from those discussed earlier, and the solutions tend to involve trigonometric functions as well as exponential functions. Here we concentrate primarily on second-order equations with constant coefficients.
- https://math.libretexts.org/Bookshelves/Calculus/Calculus_by_David_Guichard_(Improved)/17%3A_Differential_Equations/17.03%3A_First_Order_Linear_EquationsAs you might guess, a first order linear differential equation has the form y' + p(t)y = f(t). Not only is this closely related in form to the first order homogeneous linear equation, we can use what ...As you might guess, a first order linear differential equation has the form y' + p(t)y = f(t). Not only is this closely related in form to the first order homogeneous linear equation, we can use what we know about solving homogeneous equations to solve the general linear equation.
- https://math.libretexts.org/Courses/Monroe_Community_College/MTH_225_Differential_Equations/10%3A_Linear_Systems_of_Differential_Equations/10.07%3A_Variation_of_Parameters_for_Nonhomogeneous_Linear_SystemsWe now discuss an extension of the method of variation of parameters to linear nonhomogeneous systems. This method will produce a particular solution of a nonhomogenous system y′=A(t)y+f(t) provided...We now discuss an extension of the method of variation of parameters to linear nonhomogeneous systems. This method will produce a particular solution of a nonhomogenous system y′=A(t)y+f(t) provided that we know a fundamental matrix for the complementary system.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_420%3A_Differential_Equations_(Breitenbach)/10%3A_Linear_Systems_of_Differential_Equations/06%3A_Variation_of_Parameters_for_Nonhomogeneous_Linear_Systems\[\begin{array}{cccccccc}{u_{1}'}&{=}&{-\frac{1}{e^{2t}}}&{\left|\begin{array}{ccc}{e^{t}}&{e^{t}}&{e^{t}}\\{0}&{e^{t}}&{0}\\{e^{-t}}&{0}&{e^{t}}\end{array} \right|}&{=}&{-\frac{e^{3t}-e^{t}}{e^{2t}}}...\[\begin{array}{cccccccc}{u_{1}'}&{=}&{-\frac{1}{e^{2t}}}&{\left|\begin{array}{ccc}{e^{t}}&{e^{t}}&{e^{t}}\\{0}&{e^{t}}&{0}\\{e^{-t}}&{0}&{e^{t}}\end{array} \right|}&{=}&{-\frac{e^{3t}-e^{t}}{e^{2t}}}&{=}&{e^{-t}-e^{t}}\\{u_{2}'}&{=}&{-\frac{1}{e^{2t}}}&{\left|\begin{array}{ccc}{1}&{e^{t}}&{e^{t}}\\{1}&{0}&{0}\\{1}&{e^{-t}}&{e^{t}}\end{array} \right|}&{=}&{-\frac{1-e^{2t}}{e^{2t}}}&{=}&{1-e^{-2t}}\\{u_{3}'}&{=}&{-\frac{1}{e^{2t}}}&{\left|\begin{array}{ccc}{1}&{e^{t}}&{e^{t}}\\{1}&{e^{t}}&{0}\\{…
- https://math.libretexts.org/Courses/Red_Rocks_Community_College/MAT_2561_Differential_Equations_with_Engineering_Applications/05%3A_Linear_Second_Order_Equations/5.07%3A_Variation_of_ParametersThis section deals with the method traditionally called variation of parameters, which enables us to find the general solution of a nonhomogeneous linear second order equation provided that we know tw...This section deals with the method traditionally called variation of parameters, which enables us to find the general solution of a nonhomogeneous linear second order equation provided that we know two nontrivial solutions (with nonconstant ratio) of the associated homogeneous equation.
