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What is the general form of a separable differential equation?
\$\frac{dy}{dx} = g(x)h(y)$
How do you separate variables in $\frac{dy}{dx} = g(x)h(y)$?
\$\frac{dy}{h(y)} = g(x) dx$
After separating variables, what is the next step?
\$\int frac{dy}{h(y)} = int g(x) dx$
What is the general solution form after integration?
\$F(y) = G(x) + C$ where F and G are antiderivatives.
What is the integral of $\frac{1}{y}$ with respect to y?
\$\ln|y| + C$
What is the integral of $x^n$ with respect to x?
\$\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 $\ln|y| = f(x) + C$?
\$|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?
\$y = Ce^{f(x)}$ or similar, depending on the original equation.
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.
1. 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.