CSIR NET Mathematical Science UNIT – 3 Syllabus

UNIT – 3 syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for UNIT – 3 in CSIR NET Mathematical Science, not a summary of it.

1. 1

   Ordinary Differential Equations (ODEs)

   7 topics

   • Existence and uniqueness of solutions of initial value problems for first order ODEs
   • Singular solutions of first order ODEs
   • System of first order ODEs
   • General theory of homogeneous and non-homogeneous linear ODEs
   • Variation of parameters
   • Sturm-Liouville boundary value problem
   • Green’s function
2. 2

   Partial Differential Equations (PDEs)

   5 topics

   • Lagrange and Charpit methods for solving first order PDEs
   • Cauchy problem for first order PDEs
   • Classification of second order PDEs
   • General solution of higher order PDEs with constant coefficients
   • Method of separation of variables for Laplace, Heat and Wave equations
3. 3

   Numerical Analysis

   7 topics

   • Numerical solutions of algebraic equations
   • Method of iteration and Newton-Raphson method
     • Rate of convergence
   • Solution of systems of linear algebraic equations using Gauss elimination and Gauss-Seidel methods
   • Finite differences
   • Lagrange, Hermite and spline interpolation
   • Numerical differentiation and integration
   • Numerical solutions of ODEs using Picard, Euler, modified Euler and Runge-Kutta methods
4. 4

   Calculus of Variations

   3 topics

   • Variation of a functional
   • Euler-Lagrange equation
     • Necessary and sufficient conditions for extrema
   • Variational methods for boundary value problems in ordinary and partial differential equations
5. 5

   Linear Integral Equations

   4 topics

   • Linear integral equation of the first and second kind of Fredholm and Volterra type
   • Solutions with separable kernels
   • Characteristic numbers and eigenfunctions
   • Resolvent kernel
6. 6

   Classical Mechanics

   7 topics

   • Generalized coordinates
   • Lagrange’s equations
   • Hamilton’s canonical equations
   • Hamilton’s principle and principle of least action
   • Two-dimensional motion of rigid bodies
   • Euler’s dynamical equations for the motion of a rigid body about an axis
   • Theory of small oscillations

UNIT – 3 flashcards for CSIR NET Mathematical Science

25 of 50 cards from the UNIT – 3 deck — real questions with worked answers.

1. State the Picard–Lindelöf (existence and uniqueness) theorem for the first order IVP $y' = f(x,y),\; y(x_0)=y_0$.

   If $f$ is continuous in a rectangle $R$ around $(x_0,y_0)$ and satisfies a Lipschitz condition in $y$ on $R$, i.e. $|f(x,y_1)-f(x,y_2)| \leq L|y_1-y_2|$, then there exists a unique solution on some interval $|x-x_0| \leq h$.

2. What is the Lipschitz condition in $y$ for $f(x,y)$, and what easily-checked condition guarantees it?

   $|f(x,y_1)-f(x,y_2)| \leq L|y_1-y_2|$ for all relevant points, with constant $L>0$. A sufficient condition: $\frac{\partial f}{\partial y}$ exists and is bounded, $\left|\frac{\partial f}{\partial y}\right| \leq L$, on the region.

3. What does Peano's existence theorem guarantee, and how does it differ from Picard–Lindelöf?

   Peano: continuity of $f$ alone guarantees existence of at least one solution to the IVP. It does NOT guarantee uniqueness; uniqueness requires an extra condition such as Lipschitz continuity (Picard–Lindelöf).

4. Give the Picard iteration (method of successive approximations) formula for $y'=f(x,y),\,y(x_0)=y_0$.

   $$y_{n+1}(x) = y_0 + \int_{x_0}^{x} f\big(t, y_n(t)\big)\,dt,$$ starting from $y_0(x)=y_0$. The sequence $\{y_n\}$ converges uniformly to the unique solution.

5. Give an example showing failure of uniqueness when the Lipschitz condition fails.

   $y' = y^{1/3},\; y(0)=0$. Here $f=y^{1/3}$ is continuous but not Lipschitz at $y=0$. Both $y\equiv 0$ and $y=\left(\tfrac{2}{3}x\right)^{3/2}$ solve the IVP, so the solution is not unique.

6. What is a singular solution of a first order ODE, and how does it relate to the general solution?

   A singular solution is a solution not obtainable from the general solution $\phi(x,y,c)=0$ for any value of the constant $c$. Geometrically it is the envelope of the one-parameter family of general solution curves.

7. How do you find the singular solution using the $c$-discriminant and $p$-discriminant?

   $p$-discriminant: eliminate $p=y'$ from $F(x,y,p)=0$ and $\frac{\partial F}{\partial p}=0$. $c$-discriminant: eliminate $c$ from $\phi(x,y,c)=0$ and $\frac{\partial \phi}{\partial c}=0$. The singular solution (envelope) appears in both discriminant relations.

8. What is Clairaut's equation and what is its singular solution?

   Clairaut's equation: $y = px + f(p)$ where $p=y'$. General solution: $y=cx+f(c)$ (straight lines). Singular solution: obtained by eliminating $p$ from $y=px+f(p)$ and $0=x+f'(p)$; it is the envelope of those lines.

9. Write a system of first order ODEs equivalent to the $n$-th order ODE $y^{(n)}=g(x,y,y',\dots,y^{(n-1)})$.

   Set $y_1=y,\,y_2=y',\dots,y_n=y^{(n-1)}$. Then $$y_1'=y_2,\; y_2'=y_3,\;\dots,\; y_{n-1}'=y_n,\; y_n'=g(x,y_1,\dots,y_n).$$

10. For a linear system $\vec{x}' = A\vec{x}$ with constant matrix $A$, what is the general solution form, and the solution to an IVP?

    General solution $\vec{x}(t)=e^{At}\vec{c}$. With $\vec{x}(t_0)=\vec{x}_0$, the solution is $\vec{x}(t)=e^{A(t-t_0)}\vec{x}_0$. If $A$ is diagonalizable, $\vec{x}(t)=\sum_i c_i e^{\lambda_i t}\vec{v}_i$ over eigenpairs $(\lambda_i,\vec{v}_i)$.

11. Define the Wronskian of $n$ functions and state its role for solutions of a linear ODE.

    $$W(y_1,\dots,y_n)=\begin{vmatrix} y_1 & \cdots & y_n \\ y_1' & \cdots & y_n' \\ \vdots & & \vdots \\ y_1^{(n-1)} & \cdots & y_n^{(n-1)}\end{vmatrix}.$$ For solutions of an $n$-th order linear ODE, $W\neq 0$ at one point $\iff$ the solutions are linearly independent (form a fundamental set).

12. State Abel's (Liouville's) formula for the Wronskian of a second order ODE $y''+p(x)y'+q(x)y=0$.

    $$W(x) = W(x_0)\,\exp\!\left(-\int_{x_0}^{x} p(t)\,dt\right).$$ Thus $W$ is either identically zero or never zero on the interval.

13. What is the structure of the general solution of a non-homogeneous linear ODE $L[y]=g$?

    $y = y_c + y_p$, where $y_c$ (complementary function) is the general solution of the homogeneous equation $L[y]=0$, and $y_p$ is any particular solution of $L[y]=g$.

14. For the constant-coefficient ODE with characteristic root $r$ repeated $k$ times, what are the corresponding solutions? And for complex roots $\alpha\pm i\beta$?

    Repeated real root $r$ of multiplicity $k$: $e^{rx}, xe^{rx},\dots,x^{k-1}e^{rx}$. Complex conjugate roots $\alpha\pm i\beta$: $e^{\alpha x}\cos\beta x$ and $e^{\alpha x}\sin\beta x$.

15. State the variation of parameters formula for $y''+p y'+q y = g(x)$ given fundamental solutions $y_1,y_2$.

    $$y_p = -y_1\int \frac{y_2\, g}{W}\,dx + y_2\int \frac{y_1\, g}{W}\,dx,$$ where $W=W(y_1,y_2)=y_1y_2'-y_1'y_2$.

16. In variation of parameters, what equations determine $u_1'(x)$ and $u_2'(x)$ when $y_p=u_1y_1+u_2y_2$?

    $$u_1'y_1+u_2'y_2 = 0,\qquad u_1'y_1'+u_2'y_2' = g(x).$$ Solving: $u_1' = -\dfrac{y_2 g}{W},\; u_2' = \dfrac{y_1 g}{W}$.

17. Define a Sturm–Liouville (S–L) boundary value problem in standard form.

    $$\frac{d}{dx}\!\left(p(x)\frac{dy}{dx}\right) + \big(q(x) + \lambda\, w(x)\big)y = 0,\quad a\leq x\leq b,$$ with separated boundary conditions, where $w(x)>0$ is the weight function and $\lambda$ the eigenvalue parameter.

18. State the key properties of eigenvalues and eigenfunctions of a regular Sturm–Liouville problem.

    Eigenvalues are real, simple, and form an increasing sequence $\lambda_1<\lambda_2<\cdots\to\infty$. Eigenfunctions corresponding to distinct eigenvalues are orthogonal w.r.t. weight $w$: $\int_a^b y_m y_n w\,dx = 0$. The eigenfunction for $\lambda_n$ has exactly $n-1$ zeros in $(a,b)$.

19. What is the orthogonality relation for Sturm–Liouville eigenfunctions $y_m,y_n$ with weight $w(x)$?

    $$\int_a^b w(x)\,y_m(x)\,y_n(x)\,dx = 0 \quad\text{for } m\neq n.$$

20. Define Green's function $G(x,\xi)$ for the BVP $L[y]=f(x)$ with homogeneous boundary conditions, and give the solution formula.

    $G(x,\xi)$ satisfies $L[G]=\delta(x-\xi)$ with the homogeneous BCs. The solution is $$y(x)=\int_a^b G(x,\xi)\,f(\xi)\,d\xi.$$

21. List the defining properties of the Green's function for a second order Sturm–Liouville operator.

    1) $G$ satisfies the homogeneous ODE for $x\neq\xi$. 2) $G$ satisfies the boundary conditions. 3) $G$ is continuous at $x=\xi$. 4) Its derivative has a jump: $\frac{\partial G}{\partial x}\big|_{\xi^+} - \frac{\partial G}{\partial x}\big|_{\xi^-} = \frac{1}{p(\xi)}$. Also $G$ is symmetric: $G(x,\xi)=G(\xi,x)$.

22. How is Green's function constructed from two solutions $y_1,y_2$ satisfying the left and right boundary conditions?

    $$G(x,\xi)=\begin{cases} \dfrac{y_1(x)\,y_2(\xi)}{p(\xi)\,W(\xi)}, & x\leq \xi \\[2mm] \dfrac{y_1(\xi)\,y_2(x)}{p(\xi)\,W(\xi)}, & x\geq \xi \end{cases}$$ where $y_1$ meets the left BC, $y_2$ the right BC, and $W$ is their Wronskian.

23. State Lagrange's method for solving the first order linear (quasilinear) PDE $Pp + Qq = R$ where $p=z_x,\,q=z_y$.

    Form Lagrange's auxiliary (subsidiary) equations $$\frac{dx}{P}=\frac{dy}{Q}=\frac{dz}{R}.$$ Find two independent integrals $u=c_1,\,v=c_2$. The general solution is $\Phi(u,v)=0$ (or $u=\phi(v)$).

24. Outline Charpit's method for the general first order nonlinear PDE $F(x,y,z,p,q)=0$.

    Write Charpit's auxiliary equations: $$\frac{dx}{F_p}=\frac{dy}{F_q}=\frac{dz}{pF_p+qF_q}=\frac{dp}{-(F_x+pF_z)}=\frac{dq}{-(F_y+qF_z)}.$$ Find a relation $f(x,y,z,p,q,a)=0$, solve with $F=0$ for $p,q$, then integrate $dz=p\,dx+q\,dy$.

25. Write the four standard types of first order nonlinear PDEs handled by special methods (before Charpit).

    Type I: $f(p,q)=0$ (only $p,q$). Type II: $z=px+qy+f(p,q)$ (Clairaut form). Type III: $f(z,p,q)=0$ ($x,y$ absent). Type IV: $f_1(x,p)=f_2(y,q)$ (separable). Each has a complete integral $z=ax+by+c$ style ansatz.

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