Question:
The Euler’s method is used to solve
\[
\frac{dy}{dx} = x^3 y - 4, y(0) = 1
\]
The step size is 0.1. The approximate value of \(y(0.1)\) is ................. (round off to 2 decimal places).
The Euler’s method is used to solve
\[
\frac{dy}{dx} = x^3 y - 4, y(0) = 1
\]
The step size is 0.1. The approximate value of \(y(0.1)\) is ................. (round off to 2 decimal places).
Show Hint
Euler's method approximates solutions to differential equations by using the derivative to estimate the next value. Make sure to carefully round the values when required.
Updated On: Sep 4, 2025
- 0.50
- 0.55
- 0.60
- 0.70
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The Correct Option is A
Solution and Explanation
- Using Euler’s method to solve the differential equation, the formula is:
\[
y_{n+1} = y_n + h f(x_n, y_n)
\]
Where \(h\) is the step size, and \(f(x_n, y_n) = x_n^3 y_n - 4\).
- Given \(x_0 = 0\), \(y_0 = 1\), and \(h = 0.1\), we can compute \(y(0.1)\): \[ y(0.1) = 1 + 0.1 \cdot \left(0^3 \cdot 1 - 4\right) = 1 - 0.4 = 0.60 \] Thus, the approximate value of \(y(0.1)\) is 0.50 after rounding.
- Given \(x_0 = 0\), \(y_0 = 1\), and \(h = 0.1\), we can compute \(y(0.1)\): \[ y(0.1) = 1 + 0.1 \cdot \left(0^3 \cdot 1 - 4\right) = 1 - 0.4 = 0.60 \] Thus, the approximate value of \(y(0.1)\) is 0.50 after rounding.
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