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What is the formula for a particular solution given an initial condition?
$F(x) = y_0 + \int_a^x f(t) dt$, where $F(a) = y_0$ and $\frac{dy}{dx} = f(x)$.
Outline the steps to solve a separable differential equation with initial conditions.
1. Separate variables. 2. Integrate both sides. 3. Add the constant of integration (+C). 4. Use initial conditions to solve for C. 5. Substitute C back into the equation.
How do you determine the constant of integration, C, in a differential equation?
Use the given initial condition (a point (x, y)) and substitute the values into the general solution. Then, solve the equation for C.
What should you do immediately after integrating both sides of a separable differential equation?
Add the constant of integration, $+C$, to one side of the equation. This accounts for all possible solutions before applying initial conditions.
What's the first step in solving $\frac{dy}{dx} = xy$ with initial condition $y(0) = 2$?
Separate the variables: $\frac{dy}{y} = x dx$.
After separating variables and integrating, you have $\ln|y| = x^2/2 + C$. How do you solve for y?
Exponentiate both sides: $|y| = e^{x^2/2 + C} = e^C e^{x^2/2}$. Then, $y = Ae^{x^2/2}$ where $A = \pm e^C$.
Given $y = Ae^{x^2/2}$ and $y(0) = 2$, find the particular solution.
Substitute $x = 0$ and $y = 2$: $2 = Ae^{0} = A$. Thus, $A = 2$, and the particular solution is $y = 2e^{x^2/2}$.
How do you deal with an absolute value when solving differential equations?
Consider both positive and negative cases, or use the initial condition to determine the correct sign.
What do you do after finding the constant of integration?
Substitute the value of the constant back into the general solution to obtain the particular solution.
Define a general solution to a differential equation.
A family of functions that satisfies the differential equation, containing an arbitrary constant.
Define a particular solution to a differential equation.
A unique solution to a differential equation obtained by applying initial conditions to solve for the arbitrary constant in the general solution.
What is meant by 'initial condition'?
A point $(x, y)$ that a particular solution must pass through, used to determine the value of the constant $C$.
What is separation of variables?
A technique to solve differential equations by isolating each variable and its differential on opposite sides of the equation.
Define domain restriction in the context of differential equations.
Values of the independent variable for which the solution to a differential equation is not defined or valid.