What is a differential equation?

An equation containing derivatives.

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What is a differential equation?

An equation containing derivatives.

What is a separable differential equation?

A first-order differential equation where variables can be separated and integrated separately.

What is a general solution to a differential equation?

A family of functions that satisfy the differential equation.

Define 'separation of variables'.

A technique to solve differential equations by isolating variables on different sides of the equation.

What does it mean to 'solve' a differential equation?

To find a function (or a family of functions) that satisfies the equation.

What is a first-order differential equation?

A differential equation involving only first derivatives.

What are initial conditions?

Values of the function and its derivatives at a specific point, used to find a particular solution.

What is meant by 'integrating' a differential equation?

Finding the antiderivative of both sides of the separated equation to obtain a solution.

What is a 'dependent variable' in the context of differential equations?

The variable whose rate of change is being described by the differential equation (usually (y)).

What is an 'independent variable' in the context of differential equations?

The variable with respect to which the derivative is taken (usually (x)).

Solve $\frac{dy}{dx} = frac{2x}{y}$ for the general solution.

Separate: $y dy = 2x dx$ . Integrate: $\frac{1}{2}y^2 = x^2 + C$ . Solve for y: $y = \pm \sqrt{2x^2 + 2C}$ .

Solve $\frac{dy}{dx} = x^2y$ for the general solution.

Separate: $\frac{dy}{y} = x^2 dx$ . Integrate: $\ln|y| = \frac{1}{3}x^3 + C$ . Solve for y: $y = \pm Ce^{\frac{1}{3}x^3}$ .

Outline the steps to solve a separable differential equation.

Separate variables. 2. Integrate both sides. 3. Solve for y. 4. Add constant of integration.

How do you separate variables in $\frac{dy}{dx} = f(x)g(y)$ ?

Divide both sides by g(y) and multiply by dx to get $\frac{dy}{g(y)} = f(x) dx$

What is the first step in solving $\frac{dy}{dx} = xy + y$ ?

Factor out y: $\frac{dy}{dx} = y(x+1)$ .

After separating, how do you integrate $\int \frac{1}{y} dy$ ?

The integral is $\ln|y| + C$

How do you eliminate the natural logarithm in $\ln|y| = x + C$ ?

Exponentiate both sides: $e^{\ln|y|} = e^{x+C}$ , which simplifies to $|y| = e^x e^C$

How do you handle the absolute value when solving for y?

Consider both positive and negative cases: $y = \pm e^x e^C$ .

How do you simplify $e^C$ after exponentiating?

Replace $e^C$ with a new constant, often denoted as C.

What is the final form of the general solution?

The solution should be in the form $y = f(x, C)$ , where C is an arbitrary constant.

Explain the concept of separation of variables.

Rearranging a differential equation so that each variable appears on only one side, allowing for independent integration.

Why do we add a constant of integration when solving differential equations?

Because the derivative of a constant is zero, so there are infinitely many possible solutions that differ by a constant.

What does the general solution represent?

A family of curves that satisfy the differential equation. Each curve corresponds to a different value of the constant of integration.

Why is it important to check if a differential equation is separable before attempting to solve it?

Because the separation of variables method only works for separable equations, and applying it to non-separable equations will lead to incorrect results.

How does the constant of integration, C, affect the general solution?

It represents a vertical shift in the solution curve, creating a family of solutions.

What are the key steps in solving a separable differential equation?

Separate variables, integrate both sides, solve for the dependent variable, and include the constant of integration.

Explain why absolute value appears when integrating 1/y.

The integral of 1/y is ln|y|, because the natural logarithm is only defined for positive values. The absolute value ensures y can be positive or negative.

Why do we solve for y explicitly after integration?

To express the solution as a function of x, making it easier to analyze and use.

What does it mean when a differential equation is 'not separable'?

It means the terms cannot be rearranged so that each variable is isolated on one side of the equation.

How do you know if you have found the general solution?

The solution should contain an arbitrary constant and satisfy the original differential equation.

All Flashcards

What is a differential equation?

An equation containing derivatives.

What is a separable differential equation?

A first-order differential equation where variables can be separated and integrated separately.

What is a general solution to a differential equation?

A family of functions that satisfy the differential equation.

Define 'separation of variables'.

A technique to solve differential equations by isolating variables on different sides of the equation.

What does it mean to 'solve' a differential equation?

To find a function (or a family of functions) that satisfies the equation.

What is a first-order differential equation?

A differential equation involving only first derivatives.

What are initial conditions?

Values of the function and its derivatives at a specific point, used to find a particular solution.

What is meant by 'integrating' a differential equation?

Finding the antiderivative of both sides of the separated equation to obtain a solution.

What is a 'dependent variable' in the context of differential equations?

The variable whose rate of change is being described by the differential equation (usually (y)).

What is an 'independent variable' in the context of differential equations?

The variable with respect to which the derivative is taken (usually (x)).

Solve $\frac{dy}{dx} = frac{2x}{y}$ for the general solution.

Separate: $y dy = 2x dx$ . Integrate: $\frac{1}{2}y^2 = x^2 + C$ . Solve for y: $y = \pm \sqrt{2x^2 + 2C}$ .

Solve $\frac{dy}{dx} = x^2y$ for the general solution.

Separate: $\frac{dy}{y} = x^2 dx$ . Integrate: $\ln|y| = \frac{1}{3}x^3 + C$ . Solve for y: $y = \pm Ce^{\frac{1}{3}x^3}$ .

Outline the steps to solve a separable differential equation.

Separate variables. 2. Integrate both sides. 3. Solve for y. 4. Add constant of integration.

How do you separate variables in $\frac{dy}{dx} = f(x)g(y)$ ?

Divide both sides by g(y) and multiply by dx to get $\frac{dy}{g(y)} = f(x) dx$

What is the first step in solving $\frac{dy}{dx} = xy + y$ ?

Factor out y: $\frac{dy}{dx} = y(x+1)$ .

After separating, how do you integrate $\int \frac{1}{y} dy$ ?

The integral is $\ln|y| + C$

How do you eliminate the natural logarithm in $\ln|y| = x + C$ ?

Exponentiate both sides: $e^{\ln|y|} = e^{x+C}$ , which simplifies to $|y| = e^x e^C$

How do you handle the absolute value when solving for y?

Consider both positive and negative cases: $y = \pm e^x e^C$ .

How do you simplify $e^C$ after exponentiating?

Replace $e^C$ with a new constant, often denoted as C.

What is the final form of the general solution?

The solution should be in the form $y = f(x, C)$ , where C is an arbitrary constant.

Explain the concept of separation of variables.

Rearranging a differential equation so that each variable appears on only one side, allowing for independent integration.

Why do we add a constant of integration when solving differential equations?

Because the derivative of a constant is zero, so there are infinitely many possible solutions that differ by a constant.

What does the general solution represent?

A family of curves that satisfy the differential equation. Each curve corresponds to a different value of the constant of integration.

Why is it important to check if a differential equation is separable before attempting to solve it?

Because the separation of variables method only works for separable equations, and applying it to non-separable equations will lead to incorrect results.

How does the constant of integration, C, affect the general solution?

It represents a vertical shift in the solution curve, creating a family of solutions.

What are the key steps in solving a separable differential equation?

Separate variables, integrate both sides, solve for the dependent variable, and include the constant of integration.

Explain why absolute value appears when integrating 1/y.

The integral of 1/y is ln|y|, because the natural logarithm is only defined for positive values. The absolute value ensures y can be positive or negative.

Why do we solve for y explicitly after integration?

To express the solution as a function of x, making it easier to analyze and use.

What does it mean when a differential equation is 'not separable'?

It means the terms cannot be rearranged so that each variable is isolated on one side of the equation.

How do you know if you have found the general solution?

The solution should contain an arbitrary constant and satisfy the original differential equation.