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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)).
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.
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.