Question

The solution of $\frac{dy}{dx} - y - 2x + x^2 = 0$, $y(0) = 1$ at $x = 0.2$ using Euler's method is:

• A. $1.229$
• B. $1.1$
• C. $1.37$
• D. $1.521$

Hint:

When the step size $h$ is not explicitly specified in an exam question, check the options to see if they correspond to a standard subdivision like $2$ steps (e.g., $h=0.1$).

Answer 1

The Correct Option is B

Concept: Euler's method provides an iterative formula to find numerical approximations to the solution of a first-order differential equation $\frac{dy}{dx} = f(x, y)$ with an initial condition $y(x_0) = y_0$. The iterative formula is: \[ y_{n+1} = y_n + h \cdot f(x_n, y_n) \] where $h$ is the step size. Here, the differential equation can be rewritten as: \[ \frac{dy}{dx} = y + 2x - x^2 \implies f(x, y) = y + 2x - x^2 \] The given initial condition is $x_0 = 0, y_0 = 1$. We want to find the value of $y$ at $x = 0.2$. Let us assume a single direct step of size $h = 0.2$.

Step 1: Identify parameters and evaluate $f(x_0, y_0)$.
We have: \[ x_0 = 0, \quad y_0 = 1, \quad h = 0.2 \] Now, compute the value of the slope function $f(x, y)$ at the initial point $(0, 1)$: \[ f(x_0, y_0) = f(0, 1) = 1 + 2(0) - (0)^2 = 1 \]

Step 2: Apply the iteration formula to find $y_1$ at $x = 0.2$.
\[ y_1 = y_0 + h \cdot f(x_0, y_0) \] \[ y_1 = 1 + (0.2) \cdot (1) = 1 + 0.2 = 1.2 \] *(Note: If the calculation is done in two smaller steps with $h = 0.1$, let's check:
Step 1: $y(0.1) = 1 + 0.1(1) = 1.1$.
Step 2: $f(0.1, 1.1) = 1.1 + 2(0.1) - (0.1)^2 = 1.1 + 0.2 - 0.01 = 1.29$. Then $y(0.2) = 1.1 + 0.1(1.29) = 1.1 + 0.129 = 1.229$. Thus, performing the numerical approximation with 2 steps of size $h=0.1$ gives exactly $1.229$, which matches Option A perfectly!)* Let us layout the complete 2-step precise execution below: Step 1 (First Interval from $x=0$ to $x=0.1$ with $h=0.1$): \[ f(x_0, y_0) = f(0, 1) = 1 + 2(0) - 0^2 = 1 \] \[ y_1 = y_0 + h \cdot f(x_0, y_0) = 1 + 0.1 \cdot (1) = 1.1 \] Now our new coordinates are $x_1 = 0.1, y_1 = 1.1$. Step 2 (Second Interval from $x=0.1$ to $x=0.2$ with $h=0.1$): \[ f(x_1, y_1) = f(0.1, 1.1) = 1.1 + 2(0.1) - (0.1)^2 = 1.1 + 0.2 - 0.01 = 1.29 \] \[ y_2 = y_1 + h \cdot f(x_1, y_1) = 1.1 + 0.1 \cdot (1.29) = 1.1 + 0.129 = 1.229 \] Hence, the correct option matching the multi-step accurate formulation is Option 1 (1.229).