# Solve the differential equation #y'+2y=3x+1# with initial conditions #y(1) = -1# using Euler approximation ?

See below.

Making the Euler differences we have

Follow a comparison between the two Euler approximations and the exact solution of the differential equation which is

The coarser approximation is for

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To solve the differential equation (y' + 2y = 3x + 1) with initial condition (y(1) = -1) using Euler's method, follow these steps:

- Define the step size, (h).
- Initialize variables: (x_0 = 1), (y_0 = -1).
- Iterate through the following steps until reaching the desired value of (x):
- Calculate the slope at the current point: (y'(x_n) = 3x_n + 1 - 2y_n).
- Update the next (x) value: (x_{n+1} = x_n + h).
- Update the next (y) value using Euler's formula: (y_{n+1} = y_n + h \cdot y'(x_n)).

- Repeat until reaching the desired (x) value.
- The final value of (y) at the desired (x) value is the approximate solution to the differential equation.

This process approximates the solution to the given differential equation with the specified initial condition using Euler's method.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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