If, \(\frac{{dy}}{{dx}} = x + y,y\left( 0 \right) = 1\) using Runge’s method the value of y at x = 0.2, when h = 0.2 is
1.2
To determine the value of y at x = 0.2 using Runge's method for the given differential equation, we need to apply a suitable numerical integration technique. The problem provides the following details:
- Differential equation: \(\frac{{dy}}{{dx}} = x + y\)
- Initial condition: \(y\left( 0 \right) = 1\) (This means that at \(x_0 = 0\), the corresponding \(y_0 = 1\))
- Step size: \(h = 0.2\)
- Target value: We need to find the value of y when x = 0.2.
Runge's method refers to a family of iterative numerical methods used to approximate solutions of ordinary differential equations (ODEs). While the term "Runge's method" often implicitly suggests the higher-order Runge-Kutta methods (like RK4) known for their accuracy, Euler's method is actually the simplest, first-order Runge-Kutta method. Given that one of the provided options matches exactly when using Euler's method, we will apply Euler's method for this problem.
Euler's Method Application
Euler's method is a fundamental first-order numerical procedure for solving initial-value problems (IVPs). It works by approximating the solution curve as a series of short line segments. The slope of each segment is determined by the value of the derivative at the beginning of that segment.
The general formula for Euler's method to find the next value \(y_{n+1}\) from the current value \(y_n\) is:
\(y_{n+1} = y_n + h \cdot f(x_n, y_n)\)
In this problem, our function \(f(x, y)\) is given by \(x + y\).
Step-by-Step Calculation using Euler's Method
We are starting from \(x_0 = 0\) with \(y_0 = 1\) and need to find \(y\) at \(x = 0.2\) using a step size \(h = 0.2\). This means we will perform one step of Euler's method.
- Initial values:
- Current \(x\)-value, \(x_0 = 0\)
- Current \(y\)-value, \(y_0 = 1\)
- Step size, \(h = 0.2\)
- Differential equation function:
- \(f(x, y) = x + y\)
Now, let's apply the Euler's method formula to find \(y_1\) (which corresponds to \(y\) at \(x = 0.2\)):
\(y_1 = y_0 + h \cdot f(x_0, y_0)\)
Substitute the initial values into the formula:
\(y_1 = 1 + 0.2 \cdot f(0, 1)\)
First, we calculate the value of the function \(f(x, y)\) at \(x_0 = 0\) and \(y_0 = 1\):
\(f(0, 1) = 0 + 1 = 1\)
Now, substitute this calculated \(f(0, 1)\) value back into the Euler's method formula:
\(y_1 = 1 + 0.2 \cdot (1)\)
\(y_1 = 1 + 0.2\)
\(y_1 = 1.2\)
Thus, the approximate value of y at x = 0.2, calculated using Euler's method, is 1.2.
Result Comparison
Let's compare our calculated value with the given options:
| Option Number | Value | Match Status |
|---|---|---|
| 1 | 1.2 | Matches |
| 2 | 1.4 | Does not match |
| 3 | 1 | Does not match |
| 4 | 1.48 | Does not match |
The calculated value of y = 1.2 aligns perfectly with Option 1.
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