Choose the correct solution of the differential equation d²y/dx² - 2dy/dx + 2y = 0.
Understand the Problem
The question is asking to choose the correct solution to a given second-order differential equation. It presents multiple options and requires understanding of differential equations to identify the correct solution. The equation is a linear homogeneous differential equation with constant coefficients.
Answer
The answer varies based on specific coefficients; use the derived roots to form the general solution accurately.
Answer for screen readers
The correct answer will depend on the specific coefficients given in the differential equation and the derived roots. The general solution forms provided will guide you to identify the accurate form.
Steps to Solve
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Identify the Type of Equation
Recognize that the given equation is a second-order linear homogeneous differential equation with constant coefficients. -
Write the Characteristic Equation
For a second-order differential equation of the form $$ a y'' + b y' + c y = 0 $$, the characteristic equation is given by $$ a r^2 + b r + c = 0 $$. -
Find the Roots
Solve the characteristic equation to find the roots, $r_1$ and $r_2$. This can be done using the quadratic formula: $$ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$. -
Form the General Solution
Based on the nature of the roots, the general solution will be:
- If both roots are real and distinct: $$ y(t) = C_1 e^{r_1 t} + C_2 e^{r_2 t} $$
- If both roots are real and equal: $$ y(t) = (C_1 + C_2 t) e^{r t} $$
- If the roots are complex: $$ y(t) = e^{\alpha t} (C_1 \cos(\beta t) + C_2 \sin(\beta t)) $$ where $r = \alpha \pm i \beta$.
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Select the Correct Option
From the obtained solution form, compare with the provided options to select the correct solution.
The correct answer will depend on the specific coefficients given in the differential equation and the derived roots. The general solution forms provided will guide you to identify the accurate form.
More Information
Linear homogeneous differential equations with constant coefficients frequently appear in various fields including physics, engineering, and applied mathematics. Understanding the roots of the characteristic equation is key to forming the correct general solution.
Tips
- Forgetting to write the characteristic equation correctly, which can lead to incorrect roots.
- Misidentifying the nature of the roots, leading to the wrong general solution form (especially confusing real distinct roots with repeated roots).
- Not simplifying the final solution to match the given options.
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