133 lines
4.3 KiB
Markdown
133 lines
4.3 KiB
Markdown
---
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date: 09.09.2024
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type: math
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---
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## Method of Variation of Constants
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The **method of variation of constants** is a technique used to find a particular solution to a non-homogeneous linear differential equation. This method generalizes the solution of homogeneous equations by allowing the constants in the general solution to vary as functions of the independent variable.
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Notice how it's similar to [Recurrence relations](Discrete%20Structures/Recurrence%20relations.md)
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1. **Solve the homogeneous equation:** Start by solving the associated homogeneous differential equation. For an ODE of the form:
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$$
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y'' + p(x)y' + q(x)y = g(x),
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$$
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solve the homogeneous part:
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$$
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y'' + p(x)y' + q(x)y = 0.
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$$
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The general solution to the homogeneous equation will be:
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$$
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y_h(x) = C_1 y_1(x) + C_2 y_2(x),
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$$
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where $y_1(x)$ and $y_2(x)$ are linearly independent solutions.
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2. **Replace constants with functions:** Replace the constants $C_1$ and $C_2$ with functions $u_1(x)$ and $u_2(x)$:
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$$
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y_p(x) = u_1(x)y_1(x) + u_2(x)y_2(x).
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$$
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3. **Set up equations for $u_1(x)$ and $u_2(x)$:** Differentiate $y_p(x)$ and use the condition that $u_1'(x)y_1(x) + u_2'(x)y_2(x) = 0$ to avoid second derivatives of $u_1(x)$ and $u_2(x)$. This gives:
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$$
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u_1'(x)y_1(x) + u_2'(x)y_2(x) = 0,
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$$
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$$
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u_1'(x)y_1'(x) + u_2'(x)y_2'(x) = g(x).
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$$
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4. **Solve for $u_1'(x)$ and $u_2'(x)$:** Solve this system of equations to find $u_1'(x)$ and $u_2'(x)$.
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5. **Integrate to find $u_1(x)$ and $u_2(x)$:** Integrate to find $u_1(x)$ and $u_2(x)$.
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6. **Form the particular solution:** The particular solution is:
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$$
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y_p(x) = u_1(x)y_1(x) + u_2(x)y_2(x).
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$$
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## Bernoulli Equation
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A **Bernoulli equation** is a type of first-order nonlinear differential equation of the form:
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$$
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\frac{dy}{dx} + P(x)y = Q(x)y^n,
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$$
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where $n \neq 0, 1$.
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- **When and how do we apply it?**
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To solve a Bernoulli equation:
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1. **Divide through by $y^n$:**
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$$
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y^{-n} \frac{dy}{dx} + P(x)y^{1-n} = Q(x).
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$$
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2. **Make a substitution:** Let $v = y^{1-n}$. Then $\frac{dv}{dx} = (1-n)y^{-n} \frac{dy}{dx}$.
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3. **Rewrite the equation in terms of $v$:**
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$$
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\frac{dv}{dx} + (1-n)P(x)v = (1-n)Q(x).
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$$
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This is now a linear differential equation in $v(x)$.
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4. **Solve the linear ODE for $v$:**
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Use an integrating factor to solve for $v(x)$.
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5. **Substitute back to find $y(x)$:**
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Since $v = y^{1-n}$, solve for $y(x)$.
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## Riccati Equation
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A **Riccati equation** is a first-order nonlinear differential equation of the form:
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$$
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\frac{dy}{dx} = a(x) + b(x)y + c(x)y^2.
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$$
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- **When do we use it?**
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Riccati equations are used in various fields such as control theory and fluid dynamics. They can sometimes be solved by making an appropriate substitution if a particular solution is known. In general, Riccati equations do not have a straightforward general solution like linear ODEs.
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## $n \geq 2$ Linear ODE
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- **What do we do with those?**
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For linear ODEs of order $n \geq 2$, we typically look for a general solution that is a linear combination of $n$ linearly independent solutions.
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### General Properties of Spaces of Solutions of such $\epsilon$
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- **Linear dependence:** Solutions $y_1(x), y_2(x), \ldots, y_n(x)$ are linearly independent if no solution can be written as a linear combination of the others.
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- **Dimension:** The solution space of a linear homogeneous ODE of order $n$ has dimension $n$.
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- **Fundamental theorem:** If $y_1(x), y_2(x), \ldots, y_n(x)$ are $n$ linearly independent solutions to an $n$-th order linear homogeneous ODE, then any solution can be written as:
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$$
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y(x) = C_1 y_1(x) + C_2 y_2(x) + \cdots + C_n y_n(x),
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$$
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where $C_1, C_2, \ldots, C_n$ are constants.
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- **Structure of the space:** The space of solutions is a vector space, where each solution can be represented as a linear combination of a set of basis solutions.
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