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Paper Code: 41153
B-1503
M.Sc. Mathematics (Semester First) Examination, 2025
Differential Equation with Applications
Time: Three Hours ] [Maximum Marks: 70
Note: Attempt all the sections as per instructions.
Section-A
(Short Answer Type Questions)
Note: Attempt any four questions out of the follow- ing.
Each question carries 07 marks. Answer are required not exceeding 300 words.
1. Find the third approximation of the solution of the equation d/dx (y) = 2 + (- y) / x by Picord’s method where y = 2 when x = 1
2. Show that f(x, y) = x * y ^ 2 satisfies the Lipschitz condition on the rectangle R / |x| <= 1 |y| <= 1 but does not satisfy a Lipschitz condition on the strip S / |x| <= 1 |y|<00.
3. Let f,g and h be real continuous functions defined in [a,b], h(t) >= 0 for te [a,b]. Suppose on [a,b], we have the inequality f(t)≤g(t)+/h(t)f(s)ds, then prove that f(t) = g(t)+[‘h(t)g(s) exp[[“h(u)dulds in [a,b]. This is called Gronwall theorem or inequality.
4. If f(t,y) is continuous on 0 <= t <= 1 , -∞<y<00, and if |f(t, y) – f(t, x)| <= 1/t|x – y| then prove that there exists at most one solution to the initial value problem y’=f(t,y), y(0) = 0
MSc 1st Semester Maths ( Differential Equation with Applications ) Question Paper 2025 Pdf
5. For autonomous systems. Let DCIR² be a domain containing the equilibrium point of origin. If there exists a continuously differen- tiable positive definite function V:D-IR such that v = av dx ax dt ax f(x)=-w(x) is negative semi-definite in D, then prove that the equilib- rium point ‘0’ is stable.
6.Prove that the eigen values of a regular Sturm- Liouville problem are real.
7. Prove that the eigen functions of the Sturm- Liouville problem v^ prime prime + hat vv = 0 , 0 < x < I with Dirichlet boundary conditions V(0) = V(L) = 0 are orthogonal.
8. Consider the n ^ * n systern y’ * (t) =f(y(t) in the case that f(0) = 0 If a Lyapuno v function exists, the prove that the zero solution is stable.
MSc 1st Semester Maths ( Differential Equation with Applications ) Question Paper 2024-25
Section-B
(Long Answer Type Questions)
Note: Attempt any three questions out of the following.
Each question carries 14 marks. Answer required not exceeding 600-800 words.
9. State and prove Banach contraction principle.
10. Consider the IvP, y = f(x, y) and y(x_{0}) = y_{0} Let D subset I * R ^ 2 be an open set containing (x_{0} + y_{0}) Assume that for 11(x, y_{1}) (x, y_{2}) \in D | f(x, y_{1}) – f(x,y 2 )|<= phi(y_{1} – y_{2}) for some continuous function phi / [0, ∞) -> [0, ∞) such that Phi(u) > 0 for u > 0 and Phi(0) = 0 int 0 ^ 1 du phi((u)) =+ infty . Then prove that no more than passes through (x_{0}, y_{0})
11. Given a differential equation d/dt (x) = F(x) in the plane. Assume x(t) is a solution curve which stays in a bounded region. The show that either x(t) converges for t -> ∞ to an equilibrium point where F(x) = 0 or it converges to a single periodic cycle.
MSc 1st Semester Maths ( Differential Equation with Applications ) Question Paper 2024-25
12. Suppose that phi_{1} and Phi_{2} be a fundamental pair of solution (and hence are linearly independent) of y^ prime prime + q(x) * y = 0 Then prove that zero non- trival solution of this equation are isolated.
13. Find the expansion of f(x) = x in the orthonormal basis \ sqrt 1 L \ cup \{sqrt(2/L) * cos((n*pi*x)/L)\} which are eigen values of the Sturm-Liouville problem v ^ prime prime + lambda r = 0 0 < x < L v^ * (0)=v^ * (L) = 0
14. Consider the nn system y^ * (t) = f(y(t)) when f(0) = 0 If there is a strict Lyapunov function, then prove that the zero solution is asymptotically stable.
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