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2. Determine whether the given function is a solution to the given differential equation (a) x² 2y? In(y) ; y' = x² + y² (b) y e=t° dt ; y' – 2xy 1 0.

2. Determine whether the given function is a solution to the given differential equation (a) x² 2y? In(y) ; y' = x² + y² (b) y e=t° dt ; y' – 2xy 1 0.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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### Differential Equations: Verification of Solutions **Problem Statement:** Determine whether the given function is a solution to the specified differential equation. (a) Given function: \( x^2 = 2y^2 \ln(y) \) Given differential equation: \( y' = \frac{xy}{x^2 + y^2} \) (b) Given function: \( y = e^{x^2} \int_0^x e^{-t^2} \, dt \) Given differential equation: \( y' - 2xy = 1 \) #### Steps to Verify: 1. **For part (a):** - Verify if the function \( x^2 = 2y^2 \ln(y) \) satisfies the differential equation \( y' = \frac{xy}{x^2 + y^2} \). - Differentiate the function implicitly with respect to \( x \). - Simplify to see if it matches the right side of the given differential equation. 2. **For part (b):** - Verify if the function \( y = e^{x^2} \int_0^x e^{-t^2} \, dt \) satisfies the differential equation \( y' - 2xy = 1 \). - Compute the first derivative \( y' \) using the given integral form of \( y \). - Substitute \( y' \) into the differential equation. - Simplify to check if the equation holds true. By performing these steps, one can confirm whether the provided functions are indeed solutions to their respective differential equations.
Transcribed Image Text:### Differential Equations: Verification of Solutions **Problem Statement:** Determine whether the given function is a solution to the specified differential equation. (a) Given function: \( x^2 = 2y^2 \ln(y) \) Given differential equation: \( y' = \frac{xy}{x^2 + y^2} \) (b) Given function: \( y = e^{x^2} \int_0^x e^{-t^2} \, dt \) Given differential equation: \( y' - 2xy = 1 \) #### Steps to Verify: 1. **For part (a):** - Verify if the function \( x^2 = 2y^2 \ln(y) \) satisfies the differential equation \( y' = \frac{xy}{x^2 + y^2} \). - Differentiate the function implicitly with respect to \( x \). - Simplify to see if it matches the right side of the given differential equation. 2. **For part (b):** - Verify if the function \( y = e^{x^2} \int_0^x e^{-t^2} \, dt \) satisfies the differential equation \( y' - 2xy = 1 \). - Compute the first derivative \( y' \) using the given integral form of \( y \). - Substitute \( y' \) into the differential equation. - Simplify to check if the equation holds true. By performing these steps, one can confirm whether the provided functions are indeed solutions to their respective differential equations.
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