Separable Differential Equations Practice

Front

What is a separable differential equation?

Back

A separable differential equation is one that can be expressed in the form \( \frac{dy}{dx} = g(x)h(y) \), allowing the variables to be separated on opposite sides of the equation.

Front

Identify if the following differential equation is separable: \( \frac{dy}{dx} = y^2 + xy^2 \)

Back

Separable.

Front

Solve the differential equation \( \frac{dq}{dx} = \frac{3x}{q} \) given the initial condition \( q = -4 \) when \( x = 6 \).

Back

\( q = -\sqrt{3x^2 - 92} \)

Front

Solve the differential equation \( \frac{dy}{dx} = \frac{5x}{y} \) given the point (0,-1).

Back

\( y = -\sqrt{5x^2 + 1} \)

Front

How do you separate the differential equation \( \frac{dy}{dx} = \frac{2x + 1}{3y} \)?

Back

\( 3y \cdot dy = (2x + 1)dx \)

Front

What is the general solution form for a separable differential equation?

Back

The general solution form is \( y = C e^{\int g(x) dx} + h(y) \), where C is a constant.

Front

What is the first step in solving a separable differential equation?

Back

The first step is to rearrange the equation to isolate the variables, placing all terms involving y on one side and all terms involving x on the other.

Front

What does it mean for a differential equation to be non-separable?

Back

A non-separable differential equation cannot be expressed in the form \( \frac{dy}{dx} = g(x)h(y) \) and cannot be solved by separating the variables.

Front

Solve the differential equation \( \frac{dy}{dx} = x(y + 5) \).

Back

\( y = Ce^{\frac{1}{2}x^2} - 5 \)

Front

What is the significance of the initial condition in solving differential equations?

Back

The initial condition provides a specific value that allows for the determination of the constant of integration in the general solution.