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What is the general form of a separable differential equation?

<math-inline>\frac{dy}{dx} = g(x)h(y)

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What is the general form of a separable differential equation?

<math-inline>\frac{dy}{dx} = g(x)h(y)

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

<math-inline>\frac{dy}{h(y)} = g(x) dx

After separating variables, what is the next step?

<math-inline>\int frac{dy}{h(y)} = int g(x) dx

What is the general solution form after integration?

<math-inline>F(y) = G(x) + C where F and G are antiderivatives.

What is the integral of $\frac{1}{y}$ with respect to y?

<math-inline>\ln|y| + C

What is the integral of $x^n$ with respect to x?

<math-inline>\frac{x^{n+1}}{n+1} + C

How do you solve for y after integrating?

Isolate y algebraically after integration: $y = f(x, C)$

What is the exponential form of $\ln|y| = f(x) + C$ ?

<math-inline>|y| = e^{f(x) + C} = e^C e^{f(x)}

How can you simplify $e^C$ in the general solution?

Replace $e^C$ with another constant, often just $C$ .

What's the general form of the solution after simplifying constants?

<math-inline>y = Ce^{f(x)} or similar, depending on the original equation.

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 the general form of a separable differential equation?

<math-inline>\frac{dy}{dx} = g(x)h(y)

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

<math-inline>\frac{dy}{h(y)} = g(x) dx

After separating variables, what is the next step?

<math-inline>\int frac{dy}{h(y)} = int g(x) dx

What is the general solution form after integration?

<math-inline>F(y) = G(x) + C where F and G are antiderivatives.

What is the integral of $\frac{1}{y}$ with respect to y?

<math-inline>\ln|y| + C

What is the integral of $x^n$ with respect to x?

<math-inline>\frac{x^{n+1}}{n+1} + C

How do you solve for y after integrating?

Isolate y algebraically after integration: $y = f(x, C)$

What is the exponential form of $\ln|y| = f(x) + C$ ?

<math-inline>|y| = e^{f(x) + C} = e^C e^{f(x)}

How can you simplify $e^C$ in the general solution?

Replace $e^C$ with another constant, often just $C$ .

What's the general form of the solution after simplifying constants?

<math-inline>y = Ce^{f(x)} or similar, depending on the original equation.

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.