Question

What kinds of differential equations can be solved by direct integration?

Short answer

Short Answer: To solve a differential equation using direct integration, the equation must be separable, meaning it can be written in the form dy/dx = f(x) * g(y), where f(x) is a function of x only and g(y) is a function of y only. To solve the equation, we rewrite it as dy/g(y) = f(x) dx, and then we integrate both sides separately. Finally, we solve for y to find the general solution.

Step-by-step solution

Identify separable differential equations

A differential equation can be solved by direct integration if it is separable. A separable differential equation is one in which the independent variable (usually x) and dependent variable (usually y) can be separated, allowing us to integrate both sides with respect to the appropriate variable.

Write the general form of a separable differential equation

The general form of a separable differential equation is: dy/dx = f(x) * g(y) where dy/dx is the first derivative of y with respect to x, f(x) is a function of x only, and g(y) is a function of y only.

Explain the direct integration method

To solve a separable differential equation using direct integration, we can follow these steps: 1. Rewrite the equation in the form: dy/g(y) = f(x) dx 2. Integrate both sides separately: ∫1/g(y) dy = ∫f(x) dx + C 3. Solve for y to find the general solution. For example, if we have the differential equation dy/dx = x * y, we can rewrite it as dy/y = x dx. Integrating both sides, we obtain ∫1/y dy = ∫x dx + C, which yields ln|y| = 1/2x^2 + C. Solving for y, we find the general solution: y(x) = ±e^(1/2x^2 + C) or y(x) = k*e^(1/2x^2), where k = ±e^C is an arbitrary constant.