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A. Yp(x) = Ax² + Bx + C, B. Y,(x) = Ae², c. y,(x)= Acos 2x + Bsin 2z D. yp(x) = (Ax +B) cos 2x + (Cx + D) sin 2x E. Y½(x) = Axe", and F. Yp(x) = e³" (A cos 2x + B sin 2z) %3D d²y + 4y = x dz? 20 d²y dy +6 + 8y = e²z dr dr? y" + 4y + 13y = 3 cos 2z y" – 2y' – 15y = e cos 2x

A. Yp(x) = Ax² + Bx + C, B. Y,(x) = Ae², c. y,(x)= Acos 2x + Bsin 2z D. yp(x) = (Ax +B) cos 2x + (Cx + D) sin 2x E. Y½(x) = Axe", and F. Yp(x) = e³" (A cos 2x + B sin 2z) %3D d²y + 4y = x dz? 20 d²y dy +6 + 8y = e²z dr dr? y" + 4y + 13y = 3 cos 2z y" – 2y' – 15y = e cos 2x

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### Differential Equations and Particular Solutions In this section, we explore different forms of particular solutions for various differential equations, labeled from A to F. Each form is used based on the non-homogeneous part of the differential equation. #### Forms of Particular Solutions: - **A.** \( y_p(x) = Ax^2 + Bx + C \) - **B.** \( y_p(x) = Ae^{2x} \) - **C.** \( y_p(x) = A \cos 2x + B \sin 2x \) - **D.** \( y_p(x) = (Ax + B) \cos 2x + (Cx + D) \sin 2x \) - **E.** \( y_p(x) = Axe^{2x} \) - **F.** \( y_p(x) = e^{3x} (A \cos 2x + B \sin 2x) \) #### Differential Equations: Determine the form of the particular solution for each given differential equation: 1. \( \frac{d^2 y}{dx^2} + 4y = x - \frac{x^2}{20} \) 2. \( \frac{d^2 y}{dx^2} + 6 \frac{dy}{dx} + 8y = e^{2x} \) 3. \( y'' + 4y' + 13y = 3 \cos 2x \) 4. \( y'' - 2y' - 15y = e^{3x} \cos 2x \) ### Explanation: For each differential equation, the particular solution is chosen based on the form of the non-homogeneous part. For instance: - **Equation 1** involves polynomial terms, so a polynomial form like **A** may be suitable. - **Equation 2** features an exponential term \( e^{2x} \), making form **B** appropriate. - **Equation 3** involves trigonometric functions \( \cos 2x \), suggesting form **C** or **D** depending on whether additional terms are needed. - **Equation 4** combines exponential and trigonometric terms \( e^{3x} \cos 2x \), fitting form **F**. These forms assist in constructing a particular solution for each differential equation, providing
Transcribed Image Text:### Differential Equations and Particular Solutions In this section, we explore different forms of particular solutions for various differential equations, labeled from A to F. Each form is used based on the non-homogeneous part of the differential equation. #### Forms of Particular Solutions: - **A.** \( y_p(x) = Ax^2 + Bx + C \) - **B.** \( y_p(x) = Ae^{2x} \) - **C.** \( y_p(x) = A \cos 2x + B \sin 2x \) - **D.** \( y_p(x) = (Ax + B) \cos 2x + (Cx + D) \sin 2x \) - **E.** \( y_p(x) = Axe^{2x} \) - **F.** \( y_p(x) = e^{3x} (A \cos 2x + B \sin 2x) \) #### Differential Equations: Determine the form of the particular solution for each given differential equation: 1. \( \frac{d^2 y}{dx^2} + 4y = x - \frac{x^2}{20} \) 2. \( \frac{d^2 y}{dx^2} + 6 \frac{dy}{dx} + 8y = e^{2x} \) 3. \( y'' + 4y' + 13y = 3 \cos 2x \) 4. \( y'' - 2y' - 15y = e^{3x} \cos 2x \) ### Explanation: For each differential equation, the particular solution is chosen based on the form of the non-homogeneous part. For instance: - **Equation 1** involves polynomial terms, so a polynomial form like **A** may be suitable. - **Equation 2** features an exponential term \( e^{2x} \), making form **B** appropriate. - **Equation 3** involves trigonometric functions \( \cos 2x \), suggesting form **C** or **D** depending on whether additional terms are needed. - **Equation 4** combines exponential and trigonometric terms \( e^{3x} \cos 2x \), fitting form **F**. These forms assist in constructing a particular solution for each differential equation, providing
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