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- 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)/09%3A_Laplace_Transforms/9.03%3A_Solution_of_Initial_Value_Problems\[{\cal L}\left(\sin(\omega t)\right)={\cal L}\left({-1\over \omega}{d\over dt}\cos(\omega t)\right)={-1\over \omega}{\cal L}\left({d\over dt}\cos(\omega t)\right)={-1\over \omega}\left(s{\cal L}(\cos...\[{\cal L}\left(\sin(\omega t)\right)={\cal L}\left({-1\over \omega}{d\over dt}\cos(\omega t)\right)={-1\over \omega}{\cal L}\left({d\over dt}\cos(\omega t)\right)={-1\over \omega}\left(s{\cal L}(\cos(\omega t)-\cos(0)\right)={-1\over \omega}\left(s {s\over s^2+\omega^2}-1\right)={-1\over \omega} {-\omega^2\over s^2+\omega^2}={\omega\over s^2+\omega^2}.\nonumber\] \[{\cal L}(f'')=s{\cal L}(f')-f'(0)=s(s{\cal L}(f)-f(0))-f'(0)=s^2{\cal L}(f)-sf(0)-f'(0),\nonumber\]
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/01%3A_First_Order_ODEs/1.07%3A_Existence_and_Uniqueness_of_Solutions_of_Nonlinear_EquationsAlthough 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.
- https://math.libretexts.org/Courses/Reedley_College/Differential_Equations_and_Linear_Algebra_(Zook)/13%3A_Laplace_Transforms/13.03%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/Community_College_of_Denver/MAT_2562_Differential_Equations_with_Linear_Algebra/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/Chabot_College/Math_4%3A_Differential_Equations_(Dinh)/06%3A_Laplace_Transforms/6.03%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/Mission_College/Math_4B%3A_Differential_Equations_(Reed)/07%3A_Laplace_Transforms/7.03%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/Mt._San_Jacinto_College/Differential_Equations_(No_Linear_Algebra_Required)/07%3A_Laplace_Transform/7.03%3A_Solution_of_Initial_Value_Problems\[{\cal L}^{-1}(Y(s))={1\over9}{\cal L}^{-1}\Big({1\over s^2}\Big)-{1\over81}{\cal L}^{-1}\Big({1\over s}\Big)+{82\over81}{\cal L}^{-1}\Big({1\over s+9}\Big),\nonumber \] \[{\cal L}^{-1}(Y(s))=-{\cal ...\[{\cal L}^{-1}(Y(s))={1\over9}{\cal L}^{-1}\Big({1\over s^2}\Big)-{1\over81}{\cal L}^{-1}\Big({1\over s}\Big)+{82\over81}{\cal L}^{-1}\Big({1\over s+9}\Big),\nonumber \] \[{\cal L}^{-1}(Y(s))=-{\cal L}^{-1}\Big({1\over s-2}\Big)+{1\over2}{\cal L}^{-1}\Big({1\over s-5}\Big)+{5\over2}{\cal L}^{-1}\Big({1\over s-1}\Big),\nonumber \] \[{\cal L}^{-1}(Y(s))={4\over3}{\cal L}^{-1}\Big({1\over s+1/2}\Big)-8{\cal L}^{-1}\Big({1\over s+1}\Big)+{8\over3}{\cal L}^{-1}\Big({1\over s+2}\Big),\nonumber \]
- https://math.libretexts.org/Courses/Mission_College/Math_4B%3A_Differential_Equations_(Kravets)/07%3A_Laplace_Transforms/7.03%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,∞).
