Suppose that we use Euler's method to approximate the solution to the differential equation
đđŚ/đđĽ=đĽ^4/đŚ đŚ(0.1)=1
Let đ(đĽ,đŚ)=đĽ^4/đŚ.
We let đĽ0=0.1 and đŚ0=1 and pick a step size â=0.2. Euler's method is the the following algorithm. From đĽđ and đŚđ, our approximations to the solution of the differential equation at the nth stage, we find the next stage by computing
at (0.1 , 1 )
slope = dy/dx = (0.1)^4 /1 = (10^-1)^4 = 10^-4
at x = x + h = 0.1 + 0.2 = 0.3
Using Euler
our guess at y = 1 + h (dy/dx) = 1 + 0.2 * 10^-4 = 1 +0.00002 = 1.00002
the value of đŚ at the next stage using the formula:
đŚđââ = đŚđ + â * đ(đĽđ, đŚđ)
In this case, đ(đĽđ, đŚđ) = đĽđ^4 / đŚđ.
So, substituting the given values đĽ0=0.1, đŚ0=1, and â=0.2 into the Euler's method formula, we have:
đŚâ = đŚâ + â * đ(đĽâ, đŚâ)
= 1 + 0.2 * (0.1^4 / 1)
= 1 + 0.2 * (0.00001)
= 1 + 0.000002
â 1.000002
Therefore, the approximation for đŚ at the first stage is approximately 1.000002.
To find the next stage using Euler's method, follow these steps:
Step 1: Start with the initial conditions đĽ0 = 0.1 and đŚ0 = 1.
Step 2: Calculate the slope or derivative đ(đĽ, đŚ) at the current point (đĽđ, đŚđ). In this case, đ(đĽ, đŚ) = đĽ^4/đŚ.
Step 3: Use the calculated slope to estimate the value of đŚ at the next point. The slope can be approximated at the current point as đ(đĽđ, đŚđ) = (đĽđ)^4/đŚđ. The estimated next value of đŚ is given by đŚđââ = đŚđ + â * đ(đĽđ, đŚđ), where â is the step size (given as 0.2 in this case).
Step 4: Use the step size â to calculate the next value of đĽ. The next value of đĽ is given by đĽđââ = đĽđ + â.
Step 5: Repeat step 2 to step 4 with the updated values of đĽ and đŚ until you reach the desired approximation or the desired number of iterations.
Following this algorithm, you can approximate the solution to the given differential equation using Euler's method.
To use Euler's method to approximate the solution to the given differential equation, we follow these steps:
1. Start with the given initial condition đŚ(0.1) = 1, đĽ0 = 0.1, đŚ0 = 1, and the chosen step size â = 0.2.
2. Compute the slope at the current point (đĽđ, đŚđ) using the function đ(đĽ, đŚ) = đĽ^4/đŚ. In this case, we have đ(đĽđ, đŚđ) = đĽđ^4/đŚđ.
3. Use the current slope to approximate the value of the function at the next point (đĽđ+1, đŚđ+1). The approximation formula for Euler's method is:
đŚđ+1 = đŚđ + â * đ(đĽđ, đŚđ)
4. Repeat steps 2 and 3 until you reach the desired number of steps or the desired endpoint.
Using the above algorithm, we can approximate the solution to the differential equation by iteratively applying Euler's method.
For example:
- Starting with đĽ0 = 0.1 and đŚ0 = 1.
- Compute the slope at (0.1, 1) using đ(đĽ, đŚ) = đĽ^4/đŚ: đ(0.1, 1) = 0.1^4/1 = 0.00001.
- Approximate the value at the next point (đĽ1, đŚ1):
đŚ1 = 1 + 0.2 * 0.00001 = 1.000002
Repeat these steps until you reach the desired endpoint or the desired number of steps.
Note: Euler's method provides a first-order approximation, so the accuracy of the solution improves as the step size â decreases.