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Show that given function is a solution of the differential equation. Then solve the equation by finding a general solution. a) d'y 1 dy dx² x dx - 4x²y = 0, -∞

Show that given function is a solution of the differential equation. Then solve the equation by finding a general solution. a) d'y 1 dy dx² x dx - 4x²y = 0, -∞

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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Question
**Differential Equations and Solutions** 1. **Problem Statement:** Show that the given function is a solution of the differential equation. Then, solve the equation by finding a general solution. a) \(\frac{d^2 y}{dx^2} - \frac{1}{x} \frac{dy}{dx} - 4x^2 y = 0\), \(-\infty < x < \infty\), \(y_1 = e^{x^2}\) b) \(x^2 \frac{d^2 y}{dx^2} - 2x \frac{dy}{dx} + (x^2 + 2)y = 0\), \(x > 0\), \(y_1 = x \sin x\) c) \(\frac{d^2 y}{dx^2} + \frac{1}{x} \frac{dy}{dx} = 0\), \(x > 0\), \(y_1 = \ln x\) d) \(\frac{d^2 y}{dx^2} - \frac{1}{x} \frac{dy}{dx} + 4x^2 y = 0\), \(x > 0\), \(y_1 = \sin(x^2)\) e) \((1 - x^2) \frac{d^2 y}{dx^2} - 2x \frac{dy}{dx} + 2y = 0\), \(-1 < x < 1\), \(y_1 = x\) **Instructions:** - Carefully verify that each provided function \(y_1\) satisfies its respective differential equation. - Find the general solution for each differential equation.
Transcribed Image Text:**Differential Equations and Solutions** 1. **Problem Statement:** Show that the given function is a solution of the differential equation. Then, solve the equation by finding a general solution. a) \(\frac{d^2 y}{dx^2} - \frac{1}{x} \frac{dy}{dx} - 4x^2 y = 0\), \(-\infty < x < \infty\), \(y_1 = e^{x^2}\) b) \(x^2 \frac{d^2 y}{dx^2} - 2x \frac{dy}{dx} + (x^2 + 2)y = 0\), \(x > 0\), \(y_1 = x \sin x\) c) \(\frac{d^2 y}{dx^2} + \frac{1}{x} \frac{dy}{dx} = 0\), \(x > 0\), \(y_1 = \ln x\) d) \(\frac{d^2 y}{dx^2} - \frac{1}{x} \frac{dy}{dx} + 4x^2 y = 0\), \(x > 0\), \(y_1 = \sin(x^2)\) e) \((1 - x^2) \frac{d^2 y}{dx^2} - 2x \frac{dy}{dx} + 2y = 0\), \(-1 < x < 1\), \(y_1 = x\) **Instructions:** - Carefully verify that each provided function \(y_1\) satisfies its respective differential equation. - Find the general solution for each differential equation.
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