Ordinary Differential Equations

6.1 Basic Concepts of Differential Equations

Differential Equation (System): An equation (or system of equations) containing unknown functions and their derivatives.

Order of a Differential Equation (System): The order of the highest-order derivative of the unknown functions involved in the equation.

Ordinary Differential Equation (ODE): A differential equation (or system) where all unknown functions are functions of a single independent variable.

\[ F(x, \mathbf{y}, \mathbf{y}', \dots, \mathbf{y}^{(n)}) = 0 \]

Where \(\mathbf{y}(x) = (y_1(x), \dots, y_m(x))\) is a (set of) unknown function(s).

\[ \mathbf{y}^{(n)} = f(x, \mathbf{y}, \mathbf{y}', \dots, \mathbf{y}^{(n-1)}) \]

6.1.1 Solution of a Differential Equation

A solution of the differential equation \(F(x, \mathbf{y}, \mathbf{y}', \dots, \mathbf{y}^{(n)}) = 0\) is a function \(\mathbf{y} = \mathbf{y}(x)\) that satisfies the equation for all \(x\) in its domain:

\[ F(x, \mathbf{y}(x), \mathbf{y}'(x), \dots, \mathbf{y}^{(n)}(x)) = 0, \quad \forall x \]

An Initial Value Problem (also known as a Cauchy Problem) consists of a differential equation along with a set of initial conditions at a specific point \(x_0\):

\[ \begin{cases} F(x, \mathbf{y}, \mathbf{y}', \dots, \mathbf{y}^{(n)}) = 0, \\ \mathbf{y}(x_0) = \mathbf{C}_0, \\ \mathbf{y}'(x_0) = \mathbf{C}_1, \\ \vdots \\ \mathbf{y}^{(n-1)}(x_0) = \mathbf{C}_{n-1} \end{cases} \]

Where \(\mathbf{C}_0, \mathbf{C}_1, \dots, \mathbf{C}_{n-1}\) are given constant vectors. The goal is to find a specific solution \(\mathbf{y} = \mathbf{y}(x)\) that satisfies both the equation and these conditions.

For an \(n\)-th order differential equation \(F(x, y, y', \dots, y^{(n)}) = 0\), the General Solution \(y(x)\) is a family of functions that contains \(n\) arbitrary constants:

6.2 Homogeneous Differential Equations

Equations of the form \(y' = f\left(\frac{y}{x}\right)\) are known as homogeneous differential equations. They can be solved by using a substitution that transforms them into a separable equation.

Let \(u = \frac{y}{x}\), which implies: \[ \begin{aligned} y = ux \end{aligned} \]

Differentiating both sides with respect to \(x\) using the product rule, we get: \[ \begin{aligned} y' = u + x \frac{du}{dx} \end{aligned} \]

Substitute \(y'\) and \(\frac{y}{x}\) back into the original equation \(y' = f\left(\frac{y}{x}\right)\): \[ \begin{aligned} u + x \frac{du}{dx} = f(u) \end{aligned} \]

Rearrange the equation to isolate \(u\) and \(x\): \[ \begin{aligned} x \frac{du}{dx} = f(u) - u \end{aligned} \] \[ \begin{aligned} \frac{du}{f(u) - u} = \frac{dx}{x} \end{aligned} \]

Integrate both sides: \[ \begin{aligned} \int \frac{1}{f(u) - u} du = \int \frac{1}{x} dx \end{aligned} \] After solving the integral, replace \(u\) with \(\frac{y}{x}\) to obtain the final general solution.

6.3 Differential Equations of the Form

\[ y' = f\!\left( \frac{a_1 x + b_1 y + c_1} {a_2 x + b_2 y + c_2} \right). \]

6.3.1 Case 1

Assume that the linear system \[ \begin{cases} a_1 x + b_1 y + c_1 = 0, \\ a_2 x + b_2 y + c_2 = 0 \end{cases} \] has a solution \((x_0, y_0)\). This point serves as a center of symmetry of the direction field.

Translation of Variables:

Introduce the change of variables \[ \xi = x - x_0, \qquad \eta = y - y_0. \]

Then the differential equation becomes \[ \eta'_\xi = f\!\left( \frac{a_1 \xi + b_1 \eta} {a_2 \xi + b_2 \eta} \right) = f\!\left( \frac{a_1 + b_1 \dfrac{\eta}{\xi}} {a_2 + b_2 \dfrac{\eta}{\xi}} \right). \]

This is a homogeneous differential equation.

6.3.2 Case 2

Assume that the linear system \[ \begin{cases} a_1 x + b_1 y + c_1 = 0, \\ a_2 x + b_2 y + c_2 = 0 \end{cases} \] has no solution.

The differential equation can be rewritten as \[ y' = f\!\left( \alpha + \frac{\beta}{a_2 x + b_2 y + c_2} \right), \] where \(\alpha\) and \(\beta\) are constants.

Change of Variables:

Introduce the new variable \[ \xi = a_2 x + b_2 y + c_2. \]

Then \[ \xi'_x = a_2 + b_2 y'_x = a_2 + b_2 f\!\left(\alpha + \frac{\beta}{\xi}\right). \]

This is a separable differential equation.

6.4 Linear Differential Equations

\[ y' = a(x)\,y + f(x). \]

Here \(f(x)\) is the nonhomogeneous term.

If \(f(x) = 0\), the equation is called a homogeneous linear equation.

If \(f(x) \neq 0\), the equation is called a nonhomogeneous linear equation.

The homogeneous equation \[ y' = a(x)\,y \] is a separable differential equation.

Separating variables, we obtain \[ \frac{dy}{y} = a(x)\,dx \]

which implies \[ \ln |y| = \int_{x_0}^{x} a(t)\,dt + C_1. \]

Equivalently, \[ |y| = e^{C_1}\, e^{\int_{x_0}^{x} a(t)\,dt}. \]

Hence the general solution is \[ y(x) = C\, e^{\int_{x_0}^{x} a(t)\,dt}, \] where \(C\) is an arbitrary constant. This family of solutions includes the particular solution \(y = 0\).

\[ y' = a(x)\,y + f(x). \]

Assume a solution of the form \[ y(x) = C(x)\, e^{\int_{x_0}^{x} a(t)\,dt}. \]

Then \[ f(x) = y'(x) - a(x)\,y(x). \]

Substituting \(y(x)\) into the equation, we obtain \[ \begin{aligned} f(x) &= C'(x)\, e^{\int_{x_0}^{x} a(t)\,dt} + C(x)\, e^{\int_{x_0}^{x} a(t)\,dt} a(x) - a(x)\, C(x)\, e^{\int_{x_0}^{x} a(t)\,dt} \\ &= C'(x)\, e^{\int_{x_0}^{x} a(t)\,dt}. \end{aligned} \]

Hence, \[ C'(x) = f(x)\, e^{-\int_{x_0}^{x} a(t)\,dt}. \]

Integrating, \[ C(x) = C_0 + \int_{x_0}^{x} f(u)\, e^{-\int_{x_0}^{u} a(t)\,dt}\,du. \]

Finally, the general solution is \[ y(x) = C_0\, e^{\int_{x_0}^{x} a(t)\,dt} + e^{\int_{x_0}^{x} a(t)\,dt} \int_{x_0}^{x} f(u)\, e^{-\int_{x_0}^{u} a(t)\,dt}\,du. \]

6.4.1 Example

  1. \[ y' - y = e^{x}x^{2} - 2e^{2x}x + x\sin x + e^{-x}\cos x. \]

6.5 Reduction of Higher-Order Differential Equations to First-Order Systems

\[ F\!\left(x, y, y', y^{(2)}, \dots, y^{(n)}\right) = 0. \]

If the equation can be solved for the highest derivative, it can be written as \[ y^{(n)} = f\!\left(x, y, y', y^{(2)}, \dots, y^{(n-1)}\right). \]

Define \[ u_k(x) = y^{(k)}(x), \qquad k = 0,1,\dots,n-1. \]

The \(n\)-th order equation is equivalent to the system \[ \begin{cases} u_0' = u_1, \\ u_1' = u_2, \\ \ \vdots \\ F\!\left(x, u_0, u_1, \dots, u_{n-1}, u'_{n - 1}\right) = 0. \end{cases} \]

If the highest derivative is explicitly given, we obtain \[ \begin{cases} u_0' = u_1, \\ u_1' = u_2, \\ \ \vdots \\ u'_{n - 1} = f\!\left(x, u_0, u_1, \dots, u_{n-1}\right). \end{cases} \]

6.6 Linear Systems of Differential Equations with Constant Coefficients

Consider the linear system \[ \mathbf{v}' = A\,\mathbf{v}, \] where \(A\) is a constant matrix.

Let \[ \mathbf{v} = P\,\mathbf{u}, \] where \(P\) is an invertible matrix. Then \[ \mathbf{u}' = P^{-1}\mathbf{v}' = P^{-1}A\mathbf{v} = P^{-1}AP\,\mathbf{u}. \]

Hence, under the linear transformation \(\mathbf{v}=P\mathbf{u}\), the system is transformed into \[ \mathbf{u}' = \left(P^{-1}AP\right)\mathbf{u}. \]

6.7 Order Reduction of Higher-Order Differential Equations

Consider an \(n\)-th order differential equation of the form \[ F\!\left(x, y^{(k)}, y^{(k+1)}, \dots, y^{(n)}\right) = 0. \]

Introduce the new unknown function \[ u(x) = y^{(k)}(x). \]

Then the original equation is transformed into the system \[ \begin{cases} F\!\left(x, u, u', \dots, u^{(n-k)}\right) = 0, \\ y^{(k)} = u(x). \end{cases} \]

The order of the differential equation is thus reduced by \(k\).

Consider an \(n\)-th order differential equation of the form \[ F\!\left(y, y', \dots, y^{(n)}\right) = 0. \]

Introduce the substitution \[ u(y) = y'(x). \]

For any differentiable function \(w = w(y) = w(y(x))\), by the chain rule we have \[ \frac{dw}{dx} = \frac{dw}{dy}\frac{dy}{dx} = u\,\frac{dw}{dy}, \] that is, \[ \frac{d}{dx} = u\,\frac{d}{dy}. \]

Using this relation, the original equation is transformed into \[ \begin{cases} F\!\left( y,\, u,\, u\,\dfrac{du}{dy},\, \dots,\, \left(u\dfrac{d}{dy}\right)^{n-1}u \right) = 0, \\ \dfrac{dy}{dx} = u(y). \end{cases} \]

Consider the second-order linear differential equation \[ Ly = y'' - 3y' + 2y = x^2. \]

Introduce the differential operator \[ D = \frac{d}{dx}, \] then \[ L = D^2 - 3D + 2. \]

Factor the characteristic polynomial: \[ \lambda^2 - 3\lambda + 2 = (\lambda - 2)(\lambda - 1). \]

Accordingly, the differential operator can be factorized as \[ L = \left(\frac{d}{dx} - 2\right) \left(\frac{d}{dx} - 1\right). \]

Thus the equation becomes \[ \left(\frac{d}{dx} - 2\right) \left(\frac{d}{dx} - 1\right)y = x^2. \]

Define \[ u = \left(\frac{d}{dx} - 1\right)y = y' - y. \]

Then the original equation is equivalent to the system \[ \begin{cases} u' - 2u = x^2, \\ y' - y = u. \end{cases} \]

Consider the Euler-type differential equation \[ Ly = x^2 y'' - 2x y' + 2y = f(x). \]

Introduce the Euler differential operator \[ D_x = x\frac{d}{dx}. \]

Then the operator \(L\) can be written as \[ L = (D_x - \alpha)(D_x - \beta). \]

Thus, \[ Ly = \left(x\frac{d}{dx} - \alpha\right) \left(x\frac{d}{dx} - \beta\right)y = f(x). \]

Expanding the operator and comparing coefficients yields \[ \begin{cases} x - \alpha x - \beta x = -2x, \\ \alpha\beta = 2. \end{cases} \]

Solving this system gives \[ \alpha = 2, \qquad \beta = 1. \]

Define \[ u = \left(x\frac{d}{dx} - \beta\right)y = x y' - y. \]

Then the original equation is equivalent to the system \[ \begin{cases} x u' - 2u = f(x), \\ x y' - y = u. \end{cases} \]

6.8 High-Order Linear Differential Equation and First-Order Linear System

Consider the \(n\)-th order linear differential equation \[ y^{(n)} + a_{n-1}(x)y^{(n-1)} + \cdots + a_1(x)y' + a_0(x)y = f(x). \]

Introduce the state variables \[ \begin{aligned} u_0 &= y, \\ u_1 &= y', \\ &\;\;\vdots \\ u_{n-2} &= y^{(n-2)}, \\ u_{n-1} &= y^{(n-1)}. \end{aligned} \]

Then the equation can be rewritten as the first-order linear system \[ \begin{pmatrix} u_0 \\ u_1 \\ \vdots \\ u_{n-2} \\ u_{n-1} \end{pmatrix}' = \begin{pmatrix} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \\ -a_0(x) & -a_1(x) & -a_2(x) & \cdots & -a_{n-1}(x) \end{pmatrix} \begin{pmatrix} u_0 \\ u_1 \\ \vdots \\ u_{n-2} \\ u_{n-1} \end{pmatrix} + \begin{pmatrix} 0 \\ 0 \\ \vdots \\ 0 \\ f(x) \end{pmatrix}. \]

That is, the high-order linear differential equation is equivalent to a first-order linear differential system.

6.9 Superposition Principle for Linear Differential Systems

Let \(A(x)\) be a matrix-valued function.

  • Homogeneous linear system \[ \mathbf{y}' = A(x)\mathbf{y} \]

  • Nonhomogeneous linear system \[ \mathbf{y}' = A(x)\mathbf{y} + \mathbf{f}(x) \]

Here \(\mathbf{y}(x)\) is a vector-valued function and \(\mathbf{f}(x)\) is the nonhomogeneous (forcing) term.

6.10 Method of Variation of Constants (for Linear Systems)

Consider the homogeneous linear system \[ \mathbf{y}' = A(x)\mathbf{y}, \] where \(A(x)\) is an \(n \times n\) matrix-valued function.

Let \(\mathbf{y}_1(x), \ldots, \mathbf{y}_n(x)\) be \(n\) linearly independent solutions of this system. Define the fundamental matrix \[ U(x) = \bigl(\mathbf{y}_1(x), \ldots, \mathbf{y}_n(x)\bigr). \]

Then \(U(x)\) is invertible and satisfies \[ U'(x) = A(x)U(x). \]

Now consider the nonhomogeneous system \[ \mathbf{y}' = A(x)\mathbf{y} + \mathbf{f}(x). \]

Assume a solution of the form \[ \mathbf{y}(x) = U(x)\mathbf{C}(x), \] where \(\mathbf{C}(x)\) is an unknown vector-valued function.

Differentiate \(\mathbf{y}(x)\): \[ \begin{aligned} \mathbf{y}'(x) &= U'(x)\mathbf{C}(x) + U(x)\mathbf{C}'(x) \\ &= A(x)U(x)\mathbf{C}(x) + U(x)\mathbf{C}'(x) \\ &= A(x)\mathbf{y}(x) + U(x)\mathbf{C}'(x). \end{aligned} \]

Comparing with \[ \mathbf{y}' = A(x)\mathbf{y} + \mathbf{f}(x), \] we obtain \[ U(x)\mathbf{C}'(x) = \mathbf{f}(x). \]

Since \(U(x)\) is invertible, \[ \mathbf{C}'(x) = U(x)^{-1}\mathbf{f}(x). \]

Integrating, \[ \mathbf{C}(x) = \mathbf{C}_0 + \int_{x_0}^x U(t)^{-1}\mathbf{f}(t)\,dt. \]

Thus the general solution of the nonhomogeneous system is \[ \mathbf{y}(x) = U(x)\mathbf{C}_0 + U(x)\int_{x_0}^x U(t)^{-1}\mathbf{f}(t)\,dt. \]

  • \(U(x)\mathbf{C}_0\) is the general solution of the homogeneous system.
  • The integral term provides one particular solution of the nonhomogeneous system.
  • This method generalizes the variation of constants technique from scalar equations to linear systems.

Let \(\mathbf{y}_1,\mathbf{y}_2\) be two solutions of the linear differential equation \[ \mathbf{y}' = A(x)\mathbf{y} + \mathbf{f}(x), \] satisfying \[ \mathbf{y}_1(x_0) = \mathbf{y}_2(x_0). \]

Then \[ \mathbf{y}_1(x) = \mathbf{y}_2(x), \qquad \forall x. \]

Liouville’s Theorem. Let \(\mathbf{y}_1,\dots,\mathbf{y}_n\) be \(n\) solutions of the homogeneous linear system \[ \mathbf{y}' = A(x)\mathbf{y}. \]

Define the matrix \[ U(x) = \bigl(\mathbf{y}_1(x),\dots,\mathbf{y}_n(x)\bigr). \]

Then \(U(x)\) satisfies \[ \bigl(\det U(x)\bigr)' = \operatorname{tr} A(x)\,\det U(x). \]

Corollary 6.1 For any \(x\), the functions \(\mathbf{y}_1(x),\dots,\mathbf{y}_n(x)\) are linearly independent if and only if there exists a point \(x_0\) such that \(\mathbf{y}_1(x_0),\dots,\mathbf{y}_n(x_0)\) are linearly independent.