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Ordinary Differential Equations

Master ODE theory and techniques for CSIR NET — from existence theorems through dynamical systems and Sturm-Liouville problems.

Existence & Uniqueness First Order ODEs Higher Order Linear ODEs Systems of ODEs Sturm-Liouville Theory Dynamical Systems

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Existence & Uniqueness Theorems

The foundational results guaranteeing that initial value problems have solutions and when those solutions are unique.

📐 Initial Value Problems

Consider the IVP: \(\dfrac{dy}{dx} = f(x,y),\quad y(x_0) = y_0\). The central question is: under what conditions does a solution exist, and is it unique?

Definition — Lipschitz Condition A function \(f(x,y)\) satisfies a Lipschitz condition in \(y\) on a region \(D\) if there exists a constant \(L > 0\) such that \[|f(x,y_1) - f(x,y_2)| \le L|y_1 - y_2|\] for all \((x,y_1),(x,y_2) \in D\). The constant \(L\) is called the Lipschitz constant.

Theorem — Picard-Lindelof (Existence & Uniqueness) Let \(f(x,y)\) be continuous on a rectangle \(R = \{|x - x_0| \le a,\; |y - y_0| \le b\}\) and satisfy the Lipschitz condition in \(y\) on \(R\). Then the IVP \(y' = f(x,y),\; y(x_0) = y_0\) has a unique solution on \(|x - x_0| \le \min(a, b/M)\) where \(M = \sup_R |f|\).

Theorem — Peano's Existence Theorem If \(f(x,y)\) is merely continuous on \(R\), then the IVP has at least one solution. Uniqueness is not guaranteed without the Lipschitz condition.

Example

Show that \(y' = \sqrt{|y|},\; y(0) = 0\) has multiple solutions.

Solution: The function \(f(x,y)=\sqrt{|y|}\) is continuous but does not satisfy the Lipschitz condition near \(y=0\) since \(\partial f/\partial y = 1/(2\sqrt{|y|}) \to \infty\). Both \(y \equiv 0\) and \[y(x) = \begin{cases} 0 & x \le c \\ \tfrac{1}{4}(x-c)^2 & x > c \end{cases}\] are valid solutions for any \(c \ge 0\). This demonstrates that continuity alone (Peano) gives existence but not uniqueness.

🔄 Picard Iteration Method

The Picard iteration (method of successive approximations) constructs a sequence of functions converging to the solution:

\[y_0(x) = y_0, \qquad y_{n+1}(x) = y_0 + \int_{x_0}^{x} f(t, y_n(t))\,dt\]

Under the Picard-Lindelof hypotheses, \(\{y_n\}\) converges uniformly to the unique solution of the IVP.

Example

Apply Picard iteration to \(y' = y,\; y(0) = 1\).

\(y_0 = 1\), \(y_1 = 1 + \int_0^x 1\,dt = 1+x\), \(y_2 = 1 + \int_0^x (1+t)\,dt = 1 + x + \frac{x^2}{2}\), and in general \(y_n = \sum_{k=0}^{n}\frac{x^k}{k!}\). As \(n\to\infty\), this converges to \(y = e^x\).

First Order ODE Methods

Analytical techniques for solving first-order ordinary differential equations.

🧮 Separable & Exact Equations

Definition — Exact Equation The equation \(M(x,y)\,dx + N(x,y)\,dy = 0\) is exact if \(\dfrac{\partial M}{\partial y} = \dfrac{\partial N}{\partial x}\). Then there exists \(F(x,y)\) such that \(dF = M\,dx + N\,dy = 0\), giving implicit solution \(F(x,y) = C\).

Integrating factors: If the equation is not exact, multiply by an integrating factor \(\mu\). Common cases:

If \(\dfrac{1}{N}\left(\dfrac{\partial M}{\partial y} - \dfrac{\partial N}{\partial x}\right)\) depends only on \(x\), then \(\mu = e^{\int \frac{1}{N}(M_y - N_x)\,dx}\).
If \(\dfrac{1}{M}\left(\dfrac{\partial N}{\partial x} - \dfrac{\partial M}{\partial y}\right)\) depends only on \(y\), then \(\mu = e^{\int \frac{1}{M}(N_x - M_y)\,dy}\).

📈 Linear First Order & Bernoulli Equations

Theorem — Solution of Linear First Order ODE The equation \(y' + P(x)y = Q(x)\) has the general solution \[y(x) = e^{-\int P\,dx}\left(\int Q\,e^{\int P\,dx}\,dx + C\right)\]

Bernoulli equation: \(y' + P(x)y = Q(x)y^n\) with \(n \neq 0,1\). Substitution \(v = y^{1-n}\) reduces it to the linear equation \(v' + (1-n)P(x)v = (1-n)Q(x)\).

Example

Solve \(y' + \frac{1}{x}y = x^2 y^3\).

This is Bernoulli with \(n=3\). Let \(v = y^{-2}\), so \(v' = -2y^{-3}y'\). The equation becomes \(v' - \frac{2}{x}v = -2x^2\). The integrating factor is \(\mu = x^{-2}\), giving \(\frac{d}{dx}(x^{-2}v) = -2\). Hence \(v = -2x^3 + Cx^2\), and \(y^{-2} = -2x^3 + Cx^2\).

Higher Order Linear ODEs

Constant coefficient equations, the method of undetermined coefficients, and variation of parameters.

🔢 Constant Coefficient Equations

For \(a_ny^{(n)} + \cdots + a_1y' + a_0y = 0\) with constant coefficients, the characteristic equation is \[a_n r^n + \cdots + a_1 r + a_0 = 0\]

Definition — Fundamental Set Solutions corresponding to the roots:

Real distinct root \(r\): contributes \(e^{rx}\)
Repeated root \(r\) of multiplicity \(k\): contributes \(e^{rx}, xe^{rx}, \ldots, x^{k-1}e^{rx}\)
Complex roots \(\alpha \pm i\beta\): contribute \(e^{\alpha x}\cos\beta x,\; e^{\alpha x}\sin\beta x\)

Example

Solve \(y''' - 3y'' + 3y' - y = 0\).

Characteristic equation: \(r^3 - 3r^2 + 3r - 1 = (r-1)^3 = 0\). Triple root \(r=1\). General solution: \(y = (C_1 + C_2 x + C_3 x^2)e^x\).

🔧 Variation of Parameters

Theorem — Variation of Parameters For the non-homogeneous equation \(y'' + p(x)y' + q(x)y = g(x)\) with fundamental solutions \(y_1, y_2\) of the homogeneous part, a particular solution is \[y_p = -y_1\int\frac{y_2 g}{W}\,dx + y_2\int\frac{y_1 g}{W}\,dx\] where \(W = y_1 y_2' - y_2 y_1'\) is the Wronskian.

Theorem — Abel's Formula If \(y_1, y_2\) are solutions of \(y'' + p(x)y' + q(x)y = 0\), then the Wronskian satisfies \[W(x) = W(x_0)\exp\!\left(-\int_{x_0}^{x} p(t)\,dt\right)\]

Example

Find a particular solution of \(y'' + y = \sec x\).

Homogeneous solutions: \(y_1 = \cos x,\; y_2 = \sin x\); Wronskian \(W = 1\). By variation of parameters: \[y_p = -\cos x\int \sin x\sec x\,dx + \sin x\int \cos x\sec x\,dx = -\cos x\int \tan x\,dx + \sin x\int 1\,dx\] \[= \cos x \ln|\cos x| + x\sin x\]

Systems of ODEs

Matrix methods, the matrix exponential, stability analysis, and phase portraits for linear systems.

🔗 Matrix Exponential & Linear Systems

A linear system \(\mathbf{x}' = A\mathbf{x}\) with constant matrix \(A \in \mathbb{R}^{n\times n}\) has the general solution:

\[\mathbf{x}(t) = e^{At}\mathbf{x}(0), \qquad e^{At} = \sum_{k=0}^{\infty}\frac{(At)^k}{k!}\]

Definition — Matrix Exponential via Jordan Form If \(A = PJP^{-1}\) where \(J\) is the Jordan normal form, then \(e^{At} = Pe^{Jt}P^{-1}\). For a Jordan block \(J_k(\lambda)\) of size \(k\): \[e^{J_k(\lambda)t} = e^{\lambda t}\begin{pmatrix}1 & t & t^2/2! & \cdots \\ 0 & 1 & t & \cdots \\ \vdots & & \ddots & \\ 0 & \cdots & 0 & 1\end{pmatrix}\]

🌀 Stability & Phase Portraits

For \(\mathbf{x}' = A\mathbf{x}\), the stability of the origin depends on the eigenvalues \(\lambda_1, \lambda_2\) of \(A\):

Stable node: \(\lambda_1, \lambda_2 < 0\) (real, distinct)
Unstable node: \(\lambda_1, \lambda_2 > 0\) (real, distinct)
Saddle point: \(\lambda_1 < 0 < \lambda_2\) (real, opposite signs)
Stable spiral: \(\text{Re}(\lambda) < 0\) (complex conjugate pair)
Unstable spiral: \(\text{Re}(\lambda) > 0\) (complex conjugate pair)
Center: \(\text{Re}(\lambda) = 0\) (purely imaginary)

Example

Classify the equilibrium of \(\mathbf{x}' = \begin{pmatrix} -1 & 2 \\ -2 & -1 \end{pmatrix}\mathbf{x}\).

Characteristic polynomial: \(\lambda^2 + 2\lambda + 5 = 0\), giving \(\lambda = -1 \pm 2i\). Since \(\text{Re}(\lambda) = -1 < 0\), the origin is a stable spiral (spiraling inward). Trajectories spiral clockwise toward the origin.

Sturm-Liouville Theory & Eigenvalue Problems

Self-adjoint boundary value problems, orthogonality, and eigenfunction expansions.

📊 Sturm-Liouville Problems

Definition — Regular Sturm-Liouville Problem A regular Sturm-Liouville problem on \([a,b]\) is: \[\frac{d}{dx}\!\left[p(x)\frac{dy}{dx}\right] + q(x)y + \lambda w(x)y = 0\] with \(p, p', q, w\) continuous on \([a,b]\), \(p(x) > 0\), \(w(x) > 0\), and separated boundary conditions at \(x=a\) and \(x=b\).

Theorem — Properties of Sturm-Liouville Eigenvalues

The eigenvalues form a countably infinite set \(\lambda_1 < \lambda_2 < \cdots \to \infty\).
Eigenfunctions corresponding to distinct eigenvalues are orthogonal with respect to the weight function \(w(x)\): \(\int_a^b y_m(x)y_n(x)w(x)\,dx = 0\) for \(m \neq n\).
The eigenfunctions form a complete orthogonal system in \(L^2_w[a,b]\).

Example

Find the eigenvalues and eigenfunctions of \(y'' + \lambda y = 0\), \(y(0)=0\), \(y(\pi)=0\).

For \(\lambda > 0\), write \(\lambda = \mu^2\). General solution: \(y = A\cos\mu x + B\sin\mu x\). Boundary condition \(y(0)=0\) gives \(A=0\). Then \(y(\pi)=0\) requires \(\sin\mu\pi = 0\), so \(\mu = n\) for \(n=1,2,3,\ldots\). Eigenvalues: \(\lambda_n = n^2\); eigenfunctions: \(y_n = \sin(nx)\).

Dynamical Systems

Equilibria of nonlinear systems, Lyapunov stability, and linearization.

⚖️ Lyapunov Stability

Definition — Stability in the Sense of Lyapunov An equilibrium point \(\mathbf{x}^*\) of \(\mathbf{x}' = \mathbf{f}(\mathbf{x})\) is stable if for every \(\varepsilon > 0\), there exists \(\delta > 0\) such that \(\|\mathbf{x}(0) - \mathbf{x}^*\| < \delta\) implies \(\|\mathbf{x}(t) - \mathbf{x}^*\| < \varepsilon\) for all \(t \ge 0\). It is asymptotically stable if additionally \(\mathbf{x}(t) \to \mathbf{x}^*\) as \(t \to \infty\).

Theorem — Lyapunov's Direct Method Let \(\mathbf{x}^* = \mathbf{0}\) be an equilibrium of \(\mathbf{x}' = \mathbf{f}(\mathbf{x})\). If there exists a continuously differentiable function \(V(\mathbf{x})\) with \(V(\mathbf{0}) = 0\), \(V(\mathbf{x}) > 0\) for \(\mathbf{x} \neq \mathbf{0}\), and \(\dot{V}(\mathbf{x}) = \nabla V \cdot \mathbf{f}(\mathbf{x}) \le 0\), then \(\mathbf{0}\) is stable. If \(\dot{V} < 0\) for \(\mathbf{x}\neq\mathbf{0}\), then \(\mathbf{0}\) is asymptotically stable.

Theorem — Hartman-Grobman If the Jacobian \(D\mathbf{f}(\mathbf{x}^*)\) has no eigenvalues with zero real part (i.e., the equilibrium is hyperbolic), then the nonlinear system is topologically conjugate to its linearization \(\mathbf{x}' = D\mathbf{f}(\mathbf{x}^*)\mathbf{x}\) near \(\mathbf{x}^*\).

Example

Use a Lyapunov function to show that the origin is asymptotically stable for \(x' = -x + y^2,\; y' = -y\).

Try \(V(x,y) = x^2 + y^2\). Then: \[\dot{V} = 2x(-x+y^2) + 2y(-y) = -2x^2 + 2xy^2 - 2y^2\] Near the origin (for \(|x|\) small enough), the quadratic terms \(-2x^2 - 2y^2\) dominate, so \(\dot{V} < 0\) in a neighborhood of the origin. Hence the origin is asymptotically stable.

Key Takeaways

Picard-Lindelof requires Lipschitz continuity for uniqueness; Peano only guarantees existence.
The Wronskian determines linear independence of solutions (Abel's formula tracks its evolution).
For linear systems, the matrix exponential \(e^{At}\) encapsulates the complete solution.
Sturm-Liouville eigenfunctions are orthogonal and complete — the foundation for eigenfunction expansions.
Lyapunov's direct method avoids solving the system explicitly: find an appropriate energy-like function.
Hartman-Grobman allows linearization near hyperbolic equilibria of nonlinear systems.

Practice Problems

Problem 1

Verify that the IVP \(y' = 3y^{2/3},\; y(0) = 0\) has non-unique solutions. Find all solutions passing through the origin.

Show Solution ▼

\(f(y) = 3y^{2/3}\) is not Lipschitz at \(y=0\). Solutions: \(y \equiv 0\) and \(y = (x-c)^3\) for any \(c \le 0\), plus the trivial solution followed by any such branch starting at \(x = c \ge 0\).

Problem 2

Solve the exact equation \((2xy + 3)\,dx + (x^2 - 1)\,dy = 0\).

Show Solution ▼

Check: \(M_y = 2x = N_x\), so the equation is exact. \(F = \int (2xy+3)\,dx = x^2y + 3x + g(y)\). Then \(F_y = x^2 + g'(y) = x^2 - 1\), so \(g'(y)=-1\), \(g(y)=-y\). Solution: \(x^2y + 3x - y = C\).

Problem 3

Find the general solution of \(y^{(4)} - y = 0\).

Show Solution ▼

Characteristic equation: \(r^4 - 1 = (r-1)(r+1)(r^2+1)=0\). Roots: \(r = 1, -1, i, -i\). General solution: \(y = C_1 e^x + C_2 e^{-x} + C_3\cos x + C_4\sin x\).

Problem 4

Compute \(e^{At}\) for \(A = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\).

Show Solution ▼

Eigenvalues: \(\lambda = \pm i\). Using the series definition or Cayley-Hamilton: \(e^{At} = \begin{pmatrix} \cos t & \sin t \\ -\sin t & \cos t \end{pmatrix}\). This represents a rotation matrix — trajectories are circles (center equilibrium).

Problem 5

Show that \(V(x,y) = x^2 + 2y^2\) is a Lyapunov function for \(x' = -x^3,\; y' = -y - x^2y\). Determine stability.

Show Solution ▼

\(\dot{V} = 2x(-x^3) + 4y(-y - x^2y) = -2x^4 - 4y^2 - 4x^2y^2 \le 0\) with equality only at the origin. Since \(V > 0\) and \(\dot{V} < 0\) for \((x,y)\neq(0,0)\), the origin is asymptotically stable by Lyapunov's theorem.

Self-Assessment Quiz

1. Which condition guarantees uniqueness of the solution to an IVP \(y' = f(x,y)\)?

A Continuity of \(f\)

B Lipschitz condition in \(y\)

C Boundedness of \(f\)

D Monotonicity of \(f\)

2. The Wronskian of two solutions of \(y'' + p(x)y' + q(x)y = 0\) satisfies Abel's formula. If \(p(x) = 0\), then the Wronskian is:

A Always zero

B A nonzero constant

C An exponential function

D A periodic function

3. For the system \(\mathbf{x}' = A\mathbf{x}\) where \(A\) has eigenvalues \(2 \pm 3i\), the origin is:

A A stable spiral

B An unstable spiral

C A center

D A saddle point

4. In a Sturm-Liouville problem, eigenfunctions corresponding to distinct eigenvalues are:

A Equal in norm

B Linearly independent only

C Orthogonal w.r.t. weight \(w(x)\)

D Always polynomial

5. Hartman-Grobman theorem applies when the equilibrium is:

A Asymptotically stable

B Hyperbolic (no eigenvalues with zero real part)

C A center

D Isolated