GATE Mathematics Ordinary Differential Equations Syllabus

Ordinary Differential Equations syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Ordinary Differential Equations in GATE Mathematics, not a summary of it.

1. 1

   First Order Ordinary Differential Equations

   1 topic

   • Existence and Uniqueness Theorems for Initial Value Problems
2. 2

   Linear Ordinary Differential Equations of Higher Order with Constant Coefficients

   overview

   Examined as a single unit within Ordinary Differential Equations — no further topic split in the official outline.

3. 3

   Second Order Linear Ordinary Differential Equations with Variable Coefficients

   overview

   Examined as a single unit within Ordinary Differential Equations — no further topic split in the official outline.

4. 4

   Cauchy-Euler Equation

   overview

   Examined as a single unit within Ordinary Differential Equations — no further topic split in the official outline.

5. 5

   Method of Laplace Transforms for Solving Ordinary Differential Equations

   overview

   Examined as a single unit within Ordinary Differential Equations — no further topic split in the official outline.

6. 6

   Series Solutions

   2 topics

   • Power Series
   • Frobenius Method
7. 7

   Legendre and Bessel Functions and Their Orthogonal Properties

   overview

   Examined as a single unit within Ordinary Differential Equations — no further topic split in the official outline.

8. 8

   Systems of Linear First Order Ordinary Differential Equations

   overview

   Examined as a single unit within Ordinary Differential Equations — no further topic split in the official outline.

9. 9

   Sturm's Oscillation and Separation Theorems

   overview

   Examined as a single unit within Ordinary Differential Equations — no further topic split in the official outline.

10. 10

    Sturm-Liouville Eigenvalue Problems

    overview

    Examined as a single unit within Ordinary Differential Equations — no further topic split in the official outline.

11. 11

    Planar Autonomous Systems of Ordinary Differential Equations

    3 topics

    • Stability of Stationary Points for Linear Systems with Constant Coefficients
    • Linearized Stability
    • Lyapunov Functions

Ordinary Differential Equations flashcards for GATE Mathematics

21 of 50 cards from the Ordinary Differential Equations deck — real questions with worked answers.

1. State the Picard-Lindelöf (Existence and Uniqueness) Theorem for the IVP $y' = f(x,y),\ y(x_0)=y_0$.

   If $f$ is continuous on a rectangle $R$ containing $(x_0,y_0)$ and satisfies a Lipschitz condition in $y$ on $R$, then there exists $h>0$ such that the IVP has a unique solution $y(x)$ on $[x_0-h,\ x_0+h]$.

2. Define a Lipschitz condition in $y$ for a function $f(x,y)$ on a region $R$.

   There exists a constant $L>0$ (the Lipschitz constant) such that $$|f(x,y_1)-f(x,y_2)|\leq L\,|y_1-y_2|$$ for all $(x,y_1),(x,y_2)\in R$.

3. What simple sufficient condition guarantees that $f(x,y)$ satisfies a Lipschitz condition in $y$ on a convex region?

   If $\frac{\partial f}{\partial y}$ exists and is continuous (bounded) on the region, then $f$ is Lipschitz in $y$, with $L=\sup\left|\frac{\partial f}{\partial y}\right|$.

4. Which theorem guarantees existence (but not uniqueness) of a solution to $y'=f(x,y),\ y(x_0)=y_0$ using only continuity of $f$?

   Peano's Existence Theorem: if $f$ is continuous on a neighborhood of $(x_0,y_0)$, then at least one solution exists locally, but it need not be unique.

5. Give the classic example of an IVP that has a solution but not a unique one, and explain why.

   $y'=y^{1/3},\ y(0)=0$. Solutions include $y\equiv 0$ and $y=\left(\tfrac{2}{3}x\right)^{3/2}$. $f(y)=y^{1/3}$ is continuous but not Lipschitz at $y=0$ since $\frac{\partial f}{\partial y}\to\infty$ there, so uniqueness fails.

6. What is the Picard iteration (successive approximation) scheme for solving $y'=f(x,y),\ y(x_0)=y_0$?

   $$y_{n+1}(x)=y_0+\int_{x_0}^{x} f\big(t,\,y_n(t)\big)\,dt,\qquad y_0(x)=y_0,$$ which converges to the unique solution under the Lipschitz hypothesis.

7. The integral equation equivalent to the IVP $y'=f(x,y),\ y(x_0)=y_0$ is what?

   $$y(x)=y_0+\int_{x_0}^{x} f\big(t,y(t)\big)\,dt.$$ A continuous solution of this integral equation is equivalent to a solution of the IVP.

8. State the existence-uniqueness theorem for a first-order linear ODE $y'+p(x)y=q(x),\ y(x_0)=y_0$.

   If $p(x)$ and $q(x)$ are continuous on an interval $I$ containing $x_0$, then the IVP has a unique solution defined on all of $I$ (global, not just local).

9. Does the Lipschitz condition guarantee global existence of the solution on the whole interval? Explain.

   No. It guarantees only local existence and uniqueness. Solutions may blow up in finite time, e.g. $y'=y^{2},\ y(0)=1$ gives $y=\frac{1}{1-x}$, which escapes at $x=1$ despite $f$ being smooth/Lipschitz locally.

10. Define an ordinary point of the linear ODE $y''+P(x)y'+Q(x)y=0$.

    A point $x_0$ is an ordinary point if both $P(x)$ and $Q(x)$ are analytic at $x_0$ (i.e., have convergent power series expansions about $x_0$).

11. What does the power series method guarantee about solutions near an ordinary point $x_0$?

    There exist two linearly independent solutions of the form $y=\sum_{n=0}^{\infty} a_n (x-x_0)^n$, each analytic and convergent at least in the largest interval where $P$ and $Q$ are analytic.

12. Define a singular point of $y''+P(x)y'+Q(x)y=0$.

    A point $x_0$ where $P(x)$ or $Q(x)$ fails to be analytic (i.e., is not analytic at $x_0$).

13. Distinguish a regular singular point from an irregular singular point for $y''+P(x)y'+Q(x)y=0$.

    A singular point $x_0$ is regular if both $(x-x_0)P(x)$ and $(x-x_0)^2 Q(x)$ are analytic at $x_0$. If either fails to be analytic, $x_0$ is an irregular singular point.

14. In writing $y''+P(x)y'+Q(x)y=0$ in standard form, how do you obtain $P$ and $Q$ from $a_2(x)y''+a_1(x)y'+a_0(x)y=0$?

    Divide by the leading coefficient: $P(x)=\dfrac{a_1(x)}{a_2(x)}$ and $Q(x)=\dfrac{a_0(x)}{a_2(x)}$. Singular points typically occur where $a_2(x)=0$.

15. Outline the basic process of solving an ODE by the power series method at an ordinary point.

    Assume $y=\sum a_n x^n$; differentiate term by term for $y',y''$; substitute into the ODE; shift indices so all sums have the same power of $x$; collect coefficients and set each to zero to get a recurrence relation; solve for $a_n$ in terms of $a_0,a_1$.

16. For Legendre's equation $(1-x^2)y''-2xy'+n(n+1)y=0$, where are the singular points?

    At $x=\pm 1$, where the leading coefficient $1-x^2$ vanishes. $x=0$ is an ordinary point, so power series solutions about $0$ converge for $|x|<1$.

17. State the form of the solution assumed in the Frobenius method about a regular singular point $x_0=0$.

    $$y=x^{r}\sum_{n=0}^{\infty} a_n x^{n}=\sum_{n=0}^{\infty} a_n x^{n+r},\qquad a_0\neq 0,$$ where the exponent $r$ is to be determined.

18. How is the indicial equation obtained in the Frobenius method, and what does it determine?

    Substitute the Frobenius series into the ODE and equate the coefficient of the lowest power $x^{r}$ (the $a_0$ term) to zero. This gives the indicial equation, a quadratic in $r$ whose roots $r_1,r_2$ are the exponents at the singularity.

19. For a regular singular point at $0$ with $p_0=\lim_{x\to0}xP(x)$ and $q_0=\lim_{x\to0}x^2Q(x)$, write the indicial equation.

    $$r(r-1)+p_0 r + q_0 = 0,\quad\text{i.e.}\quad r^2+(p_0-1)r+q_0=0.$$

20. In the Frobenius method, what is the form of the two solutions when the indicial roots differ by a non-integer ($r_1-r_2\notin\mathbb{Z}$)?

    Two independent Frobenius series: $$y_1=x^{r_1}\sum_{n\geq0} a_n x^n,\qquad y_2=x^{r_2}\sum_{n\geq0} b_n x^n.$$

21. In the Frobenius method, what is the form of the solutions when the indicial roots are equal, $r_1=r_2=r$?

    $$y_1=x^{r}\sum_{n\geq0} a_n x^n,\qquad y_2=y_1\ln x + x^{r}\sum_{n\geq1} b_n x^n.$$ A logarithmic term always appears in the second solution.

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