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Question
Math
Posted 3 months ago
dydx=x2y+2 and f(0)=2
\frac{d y}{d x}=x-2 y+2 \text { and } f(0)=2 \text {. }


Use Euler's method with two steps of equal size to approximate f(2.4)f(2.4).
f(2.4)
f(2.4) \approx
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Answer from Sia
Posted 3 months ago
Solution by Steps
step 1
To use Euler's method, we first need to determine the step size, hh. Since we are approximating f(2.4)f(2.4) and starting at x=0x=0, with 2 steps, h=2.402=1.2h = \frac{2.4 - 0}{2} = 1.2
step 2
Now we apply the first step of Euler's method using the initial condition f(0)=2f(0)=2. We calculate the slope at the initial point using the differential equation dydx=x2y+2\frac{dy}{dx}=x-2y+2. At x=0x=0, the slope is 02(2)+2=20 - 2(2) + 2 = -2
step 3
We update the value of yy using the slope and the step size: y1=y0+hslope=2+1.2(2)=22.4=0.4y_1 = y_0 + h \cdot slope = 2 + 1.2 \cdot (-2) = 2 - 2.4 = -0.4. The new point to use for the next step is (x1,y1)=(1.2,0.4)(x_1, y_1) = (1.2, -0.4)
step 4
For the second step, we calculate the slope at (x1,y1)=(1.2,0.4)(x_1, y_1) = (1.2, -0.4) using the differential equation. The slope is 1.22(0.4)+2=1.2+0.8+2=41.2 - 2(-0.4) + 2 = 1.2 + 0.8 + 2 = 4
step 5
We update the value of yy again using the new slope: y2=y1+hslope=0.4+1.24=0.4+4.8=4.4y_2 = y_1 + h \cdot slope = -0.4 + 1.2 \cdot 4 = -0.4 + 4.8 = 4.4. The approximate value of f(2.4)f(2.4) is y2=4.4y_2 = 4.4
Answer
f(2.4)4.4f(2.4) \approx 4.4
Key Concept
Euler's Method for Approximating Solutions to Differential Equations
Explanation
Euler's method approximates the solution to a differential equation by taking a finite number of steps of a given size, using the slope of the tangent line at the beginning of each step to estimate the increase or decrease in the function's value.

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