Problem 8
Question
Use the Runge-Kutta method to approximate \(x(0.2)\) and \(y(0.2) .\) First use \(h=0.2\) and then use \(h=0.1\). Use a numerical solver and \(h=0.1\) to graph the solution in a neighborhood of \(t=0\). $$ \begin{aligned} &x^{\prime}=x+2 y \\ &y^{\prime}=4 x+3 y \\ &x(0)=1, y(0)=1 \end{aligned} $$
Step-by-Step Solution
Verified Answer
Approximate \(x(0.2)\) and \(y(0.2)\) with Runge-Kutta; refine with smaller \(h\). Plot solutions using a numerical solver.
1Step 1: Define the Differential Equations
The system of differential equations to approximate is given by: \[ x' = x + 2y \] \[ y' = 4x + 3y \] The initial conditions are \( x(0) = 1 \) and \( y(0) = 1 \). We are to approximate \( x(0.2) \) and \( y(0.2) \) using the Runge-Kutta method with \( h = 0.2 \) and \( h = 0.1 \).
2Step 2: Runge-Kutta Method for h=0.2
The Runge-Kutta method is applied as:For \( h = 0.2 \), perform the following calculations: **For \( x \):**\( k_1 = f(t, x, y) = x + 2y \)\( k_2 = f(t + \frac{h}{2}, x + \frac{h}{2}k_1, y) \)\( k_3 = f(t + \frac{h}{2}, x + \frac{h}{2}k_2, y) \)\( k_4 = f(t + h, x + hk_3, y) \)New \( x = x + \frac{h}{6}(k_1 + 2k_2 + 2k_3 + k_4) \)**For \( y \):**\( l_1 = g(t, x, y) = 4x + 3y \)\( l_2 = g(t + \frac{h}{2}, x, y + \frac{h}{2}l_1) \)\( l_3 = g(t + \frac{h}{2}, x, y + \frac{h}{2}l_2) \)\( l_4 = g(t + h, x, y + hl_3) \)New \( y = y + \frac{h}{6}(l_1 + 2l_2 + 2l_3 + l_4) \)After calculation, these give the approximate values after a single step for \( x(0.2) \) and \( y(0.2) \).
3Step 3: Runge-Kutta Method for h=0.1
Apply the same Runge-Kutta method as in Step 2 but with \( h = 0.1 \). This involves taking two steps from \( t = 0 \) to \( t = 0.2 \).For each step from \( t=0 \) to \( t=0.1 \), then from \( t=0.1 \) to \( t=0.2 \), repeat the calculations for \( k_1, k_2, k_3, k_4 \) for \( x \) and \( l_1, l_2, l_3, l_4 \) for \( y \).This will provide more accurate approximations for \( x(0.2) \) and \( y(0.2) \).
4Step 4: Numerical Solver and Graphing with h=0.1
Use a numerical solver (such as Python's SciPy library) to solve the system using smaller steps in a neighborhood around \( t = 0 \). Start with initial conditions \( x(0) = 1 \) and \( y(0) = 1 \).The solver will provide values for \( x(t) \) and \( y(t) \) over a range including \( t=0.2 \).Plot \( x(t) \) and \( y(t) \) against \( t \), ensuring that the graph covers a neighborhood around \( t = 0 \) to visualize the solution's behavior.
Key Concepts
Numerical ApproximationDifferential EquationsInitial Value Problems
Numerical Approximation
Numerical approximation methods are techniques used to find approximate solutions to mathematical problems that cannot be solved analytically. In the context of differential equations, these methods allow us to estimate the behavior of the system at discrete points. One commonly used technique is the Runge-Kutta method, which provides a systematic approach to approximate solutions over a series of small steps. This method is particularly helpful when the differential equations do not have a simple formula for direct solutions.
The Runge-Kutta method works by calculating intermediate points between the initial point and the point for which we want to find the solution. This involves evaluating the equation at several points between the steps, using these intermediate values to get a more accurate approximation.
This stepping approach ensures that larger problems can be tackled by dividing them into smaller, more manageable segments. The precision of our approximation depends on the size of the step, denoted by the variable \( h \). Smaller values of \( h \) result in more accurate solutions because they capture more intermediate behavior between the start and end points.
The Runge-Kutta method works by calculating intermediate points between the initial point and the point for which we want to find the solution. This involves evaluating the equation at several points between the steps, using these intermediate values to get a more accurate approximation.
This stepping approach ensures that larger problems can be tackled by dividing them into smaller, more manageable segments. The precision of our approximation depends on the size of the step, denoted by the variable \( h \). Smaller values of \( h \) result in more accurate solutions because they capture more intermediate behavior between the start and end points.
Differential Equations
Differential equations are mathematical equations that relate a function to its derivatives. They play a crucial role in modeling dynamics and changes across various systems such as physics, biology, and economics. These equations describe how a quantity changes with respect to another, often time.
For instance, in our exercise, we have two differential equations: \ 1. \( x' = x + 2y \) \ 2. \( y' = 4x + 3y \)
These equations represent a system of first-order differential equations. The prime symbol (\( x' \) and \( y' \)) indicates the derivative of \( x \) and \( y \) with respect to another variable, typically time (\( t \)). This system shows how the rate of change in \( x \) and \( y \) depends on their current values.
Solving such systems often requires initial conditions. In our case, we start with \( x(0) = 1 \) and \( y(0) = 1 \). These values provide a specific starting point for the equations, enabling us to predict future values like \( x(0.2) \) and \( y(0.2) \).
For instance, in our exercise, we have two differential equations: \ 1. \( x' = x + 2y \) \ 2. \( y' = 4x + 3y \)
These equations represent a system of first-order differential equations. The prime symbol (\( x' \) and \( y' \)) indicates the derivative of \( x \) and \( y \) with respect to another variable, typically time (\( t \)). This system shows how the rate of change in \( x \) and \( y \) depends on their current values.
Solving such systems often requires initial conditions. In our case, we start with \( x(0) = 1 \) and \( y(0) = 1 \). These values provide a specific starting point for the equations, enabling us to predict future values like \( x(0.2) \) and \( y(0.2) \).
Initial Value Problems
An initial value problem is a type of differential equation accompanied by specific starting conditions. In the case of our exercise, the initial values are \( x(0) = 1 \) and \( y(0) = 1 \). These initial conditions are essential because they specify the exact solution path among infinite possibilities that satisfy the differential equations.
Solving an initial value problem involves using these given points to find subsequent values of the functions for future times, such as \( x(0.2) \) and \( y(0.2) \). Without initial values, solving a system of differential equations becomes challenging because there are no constraints to limit the solution paths.
In applications, initial value problems are critical. They are utilized in engineering to model the spread of heat, in biology to predict population dynamics, and many other fields where prediction of future states based on present conditions is necessary. The initial value provides a baseline from which predictions are made, using methods such as the Runge-Kutta to ensure accuracy in approximation.
Solving an initial value problem involves using these given points to find subsequent values of the functions for future times, such as \( x(0.2) \) and \( y(0.2) \). Without initial values, solving a system of differential equations becomes challenging because there are no constraints to limit the solution paths.
In applications, initial value problems are critical. They are utilized in engineering to model the spread of heat, in biology to predict population dynamics, and many other fields where prediction of future states based on present conditions is necessary. The initial value provides a baseline from which predictions are made, using methods such as the Runge-Kutta to ensure accuracy in approximation.
Other exercises in this chapter
Problem 8
Use the \(\mathrm{RK} 4\) method with \(h=0.1\) to obtain a four-decimal approximation to the indicated value. $$ y^{\prime}=x+y^{2}, \quad y(0)=0 ; y(0.5) $$
View solution Problem 8
Use the Adams-Bashforth-Moulton method to approximate \(y(1.0)\), where \(y(x)\) is the solution of the given initial-value problem. First use \(h=0.2\) and the
View solution Problem 8
Given the initial-value problems, use the improved Euler's method to obtain a four-decimal approximation to the indicated value. First use \(h=0.1\) and then us
View solution Problem 8
Use the finite difference method with \(n=10\) to approximate the solution of the boundary-value problem \(y^{\prime \prime}+6.55(1+x) y=1, y(0)=0, y(1)=0\).
View solution