General Solution of Differential Equations - First Order Linear and Separable

Math Problem Statement

Das sind Aufgaben für Mathe, rechne die Aufgaben ausführlich mit lösungsweg. Unter den aufgaben sind die Lösungen bereits die raus kommen sollen.

Bestimmen Sie die allgemeine Lösung der Differenzialgleichungen: x^2 y' + 4x^3 y = 3x^3, x > 0; e^x y' - y^2 = 0

Solution

Let's walk through the solutions for these differential equation problems.

1. Bestimmen Sie die allgemeine Lösung der Differenzialgleichungen:

a) $x^2 y' + 4x^3 y = 3x^3$ , $x > 0$

The equation is first-order linear.

Step 1: Rewrite the equation
$y' + 4x \cdot y = \frac{3}{x}$ This is

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Math Problem Analysis

Mathematical Concepts

Differential Equations
Linear First-Order Equations
Homogeneous Equations

Formulas

General solution for a linear first-order differential equation: y(x) = ce^(integral P(x)) + f(x)
General solution for a separable differential equation: y(x) = F(x) based on separation of variables

Theorems

Existence and Uniqueness Theorem for Ordinary Differential Equations
Method of Integrating Factor for Linear First-Order Differential Equations

Suitable Grade Level

University level

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