Question

Use Euler's method and the Euler semilinear method with step sizes $h=0.1,h=0.05,$ and $h=0.025$ to find approximate values of the solution of the initial value problem $$y^{\prime }+3y=7e^{-3x},y(0)=6$$ at $x=0,0.1,0.2,0.3,\dots ,1.0$. Compare these approximate values with the values of the exact solution $y=e^{-3x}(7x+6)$, which can be obtained by the method of Section 2.1. Do you notice anything special about the results? Explain.

Short answer

Analyze and explain any special observations. Answer: After implementing the Euler's method and Euler semilinear method with various step sizes, we can compare the approximated values with the exact solution to assess their accuracy. In general, we may observe that as the step size decreases, the approximations become more accurate. For both methods, smaller step sizes lead to better approximations of the exact solution. However, the Euler semilinear method may provide relatively better approximations as it accounts for the linearity of the problem. The analysis should also include any specific observations regarding the differences in accuracy between the methods and possible reasons behind such outcome.

Step-by-step solution

Identify the differential equation and initial condition

The problem gives us a first-order linear differential equation and an initial condition: $y^{\prime }+3y=7e^{-3x}$ , with $y(0)=6$.

Determine the exact solution

The exact solution is given by the problem: $y(x)=e^{-3x}(7x+6)$.

Implement Euler's method

We will implement Euler's method for the step sizes $h=0.1,0.05,0.025$. The equation for Euler's method is: $y_{n+1}=y_n+hf(x_n,y_n)$ In our case, $f(x,y)=7e^{-3x}-3y$.

Calculate approximate values using different step sizes

For each step size $h$, we will calculate the values of $y(x)$ at $x=0,0.1,0.2,\dots ,1$ using Euler's method and compare it to the exact solution.

Implement Euler semilinear method

We will implement Euler semilinear method for the step sizes $h=0.1,0.05,0.025$. The equation for the Euler semilinear method is: $y_{n+1}=\frac{y_n+hP(x_n)}{1-hQ(x_n)}$ In our case, $P(x)=7e^{-3x}$ and $Q(x)=3$.

Calculate approximate values using different step sizes

For each step size $h$, we will calculate the values of $y(x)$ at $x=0,0.1,0.2,\dots ,1$ using Euler semilinear method and compare it to the exact solution.

Compare and analyze the results

Compare the approximate values obtained by Euler's method and Euler semilinear method with the exact solution at each $x$. Analyze and explain any special observations about the results.