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- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_420%3A_Differential_Equations_(Breitenbach)/03%3A_Numerical_Methods/3.03%3A_The_Runge-Kutta_Method\[\begin{aligned} k_{10} & = f(x_0,y_0) = f(0,1)=-2,\\ k_{20} & = f(x_0+h/2,y_0+hk_{10}/2)=f(.05,1+(.05)(-2))\\ &= f(.05,.9)=-2(.9)+(.05)^3e^{-.1}=-1.799886895,\\ k_{30} & = f(x_0+h/2,y_0+hk_{20}/2)=f...\[\begin{aligned} k_{10} & = f(x_0,y_0) = f(0,1)=-2,\\ k_{20} & = f(x_0+h/2,y_0+hk_{10}/2)=f(.05,1+(.05)(-2))\\ &= f(.05,.9)=-2(.9)+(.05)^3e^{-.1}=-1.799886895,\\ k_{30} & = f(x_0+h/2,y_0+hk_{20}/2)=f(.05,1+(.05)(-1.799886895))\\ &= f(.05,.910005655)=-2(.910005655)+(.05)^3e^{-.1}=-1.819898206,\\ k_{40} & = f(x_0+h,y_0+hk_{30})=f(.1,1+(.1)(-1.819898206))\\ &=f(.1,.818010179)=-2(.818010179)+(.1)^3e^{-.2}=-1.635201628,\\ y_1&=y_0+{h\over6}(k_{10}+2k_{20}+2k_{30}+k_{40}),\\ &=1+{.1\over6}(-2+2(-1.7…
- https://math.libretexts.org/Courses/Monroe_Community_College/MTH_225_Differential_Equations/03%3A_Numerical_Methods/3.03%3A_The_Runge-Kutta_Method/3.3.01%3A_The_Runge-Kutta_Method_(Exercises)In Exercises 3.3.20–3.3.22 use the Runge-Kutta method and the Runge-Kutta semilinear method with the indicated step sizes to find approximate values of the solution of the given initial value problem ...In Exercises 3.3.20–3.3.22 use the Runge-Kutta method and the Runge-Kutta semilinear method with the indicated step sizes to find approximate values of the solution of the given initial value problem at 11 equally spaced points (including the endpoints) in the interval.
- https://math.libretexts.org/Bookshelves/Applied_Mathematics/Numerical_Methods_(Chasnov)/07%3A_Ordinary_Differential_Equations/7.02%3A_Numerical_Methods_-_Initial_Value_ProblemSubstituting in \(k_{1}\) and \(k_{2}\), we have \[x_{n+1}=x_{n}+a \Delta t f\left(t_{n}, x_{n}\right)+b \Delta t f\left(t_{n}+\alpha \Delta t, x_{n}+\beta \Delta t f\left(t_{n}, x_{n}\right)\right) ....Substituting in \(k_{1}\) and \(k_{2}\), we have \[x_{n+1}=x_{n}+a \Delta t f\left(t_{n}, x_{n}\right)+b \Delta t f\left(t_{n}+\alpha \Delta t, x_{n}+\beta \Delta t f\left(t_{n}, x_{n}\right)\right) . \nonumber \] We Taylor series expand using \[\begin{aligned} f\left(t_{n}+\alpha \Delta t, x_{n}+\beta \Delta t f\left(t_{n}, x_{n}\right)\right) & \\ =f\left(t_{n}, x_{n}\right)+\alpha \Delta t f_{t}\left(t_{n}, x_{n}\right)+\beta \Delta t f\left(t_{n}, x_{n}\right) f_{x}\left(t_{n}, x_{n}\right)…
- https://math.libretexts.org/Courses/Community_College_of_Denver/MAT_2562_Differential_Equations_with_Linear_Algebra/03%3A_Numerical_Methods/3.03%3A_The_Runge-Kutta_MethodThis section deals with the Runge-Kutta method, perhaps the most widely used method for numerical solution of differential equations.
- https://math.libretexts.org/Bookshelves/Differential_Equations/Elementary_Differential_Equations_with_Boundary_Value_Problems_(Trench)/03%3A_Numerical_Methods/3.03%3A_The_Runge-Kutta_Method/3.3E%3A_The_Runge-Kutta_Method_(Exercises)In Exercises 3.3.20–3.3.22 use the Runge-Kutta method and the Runge-Kutta semilinear method with the indicated step sizes to find approximate values of the solution of the given initial value problem ...In Exercises 3.3.20–3.3.22 use the Runge-Kutta method and the Runge-Kutta semilinear method with the indicated step sizes to find approximate values of the solution of the given initial value problem at 11 equally spaced points (including the endpoints) in the interval.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_420%3A_Differential_Equations_(Breitenbach)/03%3A_Numerical_Methods/3.03%3A_The_Runge-Kutta_Method/3.3E%3A_The_Runge-Kutta_Method_(Exercises)In Exercises 1-5 use the Runge-Kutta method to find approximate values of the solution of the given initial value problem at the points \(x_i=x_0+ih\), where \(x_0\) is the point where the initial con...In Exercises 1-5 use the Runge-Kutta method to find approximate values of the solution of the given initial value problem at the points \(x_i=x_0+ih\), where \(x_0\) is the point where the initial condition is imposed and \(i=1\), \(2\). In each exercise, use the Runge-Kutta method with the indicated step size to find approximate values of the solution of the given initial value problem at 11 equally spaced points (including the endpoints) in the interval.
- https://math.libretexts.org/Courses/Community_College_of_Denver/MAT_2562_Differential_Equations_with_Linear_Algebra/03%3A_Numerical_Methods/3.03%3A_The_Runge-Kutta_Method/3.3E%3A_The_Runge-Kutta_Method_(Exercises)In Exercises 3.3.20–3.3.22 use the Runge-Kutta method and the Runge-Kutta semilinear method with the indicated step sizes to find approximate values of the solution of the given initial value problem ...In Exercises 3.3.20–3.3.22 use the Runge-Kutta method and the Runge-Kutta semilinear method with the indicated step sizes to find approximate values of the solution of the given initial value problem at 11 equally spaced points (including the endpoints) in the interval.
- https://math.libretexts.org/Courses/Red_Rocks_Community_College/MAT_2561_Differential_Equations_with_Engineering_Applications/03%3A_Numerical_Methods/3.03%3A_The_Runge-Kutta_Method/3.3.01%3A_The_Runge-Kutta_Method_(Exercises)In Exercises 3.3.20–3.3.22 use the Runge-Kutta method and the Runge-Kutta semilinear method with the indicated step sizes to find approximate values of the solution of the given initial value problem ...In Exercises 3.3.20–3.3.22 use the Runge-Kutta method and the Runge-Kutta semilinear method with the indicated step sizes to find approximate values of the solution of the given initial value problem at 11 equally spaced points (including the endpoints) in the interval.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/01%3A_First_Order_ODEs/1.10%3A_Numerical_Methods_-_Eulers_MethodThis page elaborates on Euler's method for approximating solutions to differential equations when closed-form solutions are not feasible. It discusses the method's iterative approach and its first-ord...This page elaborates on Euler's method for approximating solutions to differential equations when closed-form solutions are not feasible. It discusses the method's iterative approach and its first-order accuracy, noting the error reduction with smaller step sizes. The text also addresses challenges like numerical instability and the importance of selecting appropriate methods and step sizes.
- https://math.libretexts.org/Bookshelves/Differential_Equations/A_First_Course_in_Differential_Equations_for_Scientists_and_Engineers_(Herman)/03%3A_Numerical_Solutions/3.04%3A_Runge-Kutta_MethodsIn this section we will find approximations of to solutions that avoid the need for computing the derivatives.
- https://math.libretexts.org/Bookshelves/Differential_Equations/Elementary_Differential_Equations_with_Boundary_Value_Problems_(Trench)/03%3A_Numerical_Methods/3.03%3A_The_Runge-Kutta_MethodThis section deals with the Runge-Kutta method, perhaps the most widely used method for numerical solution of differential equations.
