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Calculus 2/Linear ODEs.md

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+---
+date: 09.09.2024
+type: math
+---
+
+## Method of Variation of Constants
+![Variation of parameters](https://www.youtube.com/watch?v=Ik3YW1JGr_A&pp=ygUgTWV0aG9kIG9mIFZhcmlhdGlvbiBvZiBDb25zdGFudHM%3D)
+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. 
+
+Notice how it's similar to [Recurrence relations](Discrete%20Structures/Recurrence%20relations.md) 
+
+
+1. **Solve the homogeneous equation:** Start by solving the associated homogeneous differential equation. For an ODE of the form:
+
+   $$
+   y'' + p(x)y' + q(x)y = g(x),
+   $$
+
+   solve the homogeneous part:
+
+   $$
+   y'' + p(x)y' + q(x)y = 0.
+   $$
+
+   The general solution to the homogeneous equation will be:
+
+   $$
+   y_h(x) = C_1 y_1(x) + C_2 y_2(x),
+   $$
+
+   where $y_1(x)$ and $y_2(x)$ are linearly independent solutions.
+
+2. **Replace constants with functions:** Replace the constants $C_1$ and $C_2$ with functions $u_1(x)$ and $u_2(x)$:
+
+   $$
+   y_p(x) = u_1(x)y_1(x) + u_2(x)y_2(x).
+   $$
+
+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:
+
+   $$
+   u_1'(x)y_1(x) + u_2'(x)y_2(x) = 0,
+   $$
+   $$
+   u_1'(x)y_1'(x) + u_2'(x)y_2'(x) = g(x).
+   $$
+
+4. **Solve for $u_1'(x)$ and $u_2'(x)$:** Solve this system of equations to find $u_1'(x)$ and $u_2'(x)$.
+
+5. **Integrate to find $u_1(x)$ and $u_2(x)$:** Integrate to find $u_1(x)$ and $u_2(x)$.
+
+6. **Form the particular solution:** The particular solution is:
+
+   $$
+   y_p(x) = u_1(x)y_1(x) + u_2(x)y_2(x).
+   $$
+
+## Bernoulli Equation
+![Understanding the bernoulli equation](https://www.youtube.com/watch?v=DW4rItB20h4)
+![Using the bernoulli equation](https://www.youtube.com/watch?v=iCN8nGXE29o)
+
+A **Bernoulli equation** is a type of first-order nonlinear differential equation of the form:
+
+$$
+\frac{dy}{dx} + P(x)y = Q(x)y^n,
+$$
+
+where $n \neq 0, 1$.
+
+- **When and how do we apply it?**
+
+To solve a Bernoulli equation:
+
+1. **Divide through by $y^n$:**
+
+   $$
+   y^{-n} \frac{dy}{dx} + P(x)y^{1-n} = Q(x).
+   $$
+
+2. **Make a substitution:** Let $v = y^{1-n}$. Then $\frac{dv}{dx} = (1-n)y^{-n} \frac{dy}{dx}$.
+
+3. **Rewrite the equation in terms of $v$:**
+
+   $$
+   \frac{dv}{dx} + (1-n)P(x)v = (1-n)Q(x).
+   $$
+
+   This is now a linear differential equation in $v(x)$.
+
+4. **Solve the linear ODE for $v$:**
+
+   Use an integrating factor to solve for $v(x)$.
+
+5. **Substitute back to find $y(x)$:**
+
+   Since $v = y^{1-n}$, solve for $y(x)$.
+
+## Riccati Equation
+![Explanation](https://www.youtube.com/watch?v=MoO7Jw06_PM)
+
+A **Riccati equation** is a first-order nonlinear differential equation of the form:
+
+$$
+\frac{dy}{dx} = a(x) + b(x)y + c(x)y^2.
+$$
+
+- **When do we use it?**
+
+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.
+
+## $n \geq 2$ Linear ODE
+
+![Constant coeff DEs](https://www.youtube.com/watch?v=is0F0u62IbY)
+- **What do we do with those?**
+
+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.
+
+### General Properties of Spaces of Solutions of such $\epsilon$
+
+- **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.
+
+- **Dimension:** The solution space of a linear homogeneous ODE of order $n$ has dimension $n$.
+
+- **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:
+
+  $$
+  y(x) = C_1 y_1(x) + C_2 y_2(x) + \cdots + C_n y_n(x),
+  $$
+
+  where $C_1, C_2, \ldots, C_n$ are constants.
+
+- **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.