PSG COLLEGE OF TECHNOLOGY, COIMBATORE - 641 004
DEPARTMENT OF MATHEMATICS
23A/C/M/P/Y/L/E/U/D/R/Z/I/N/H/T/B101- CALCULUS AND ITS APPLICATIONS
(FOR ALL BE/BTECH –I SEMESTER)
PROBLEM SHEET ON CO4 - SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS
Part – A (MCQ/Fill ups)
1. _________ is the solution of the homogeneous linear ODE y' ' y 0
A) 𝑦 = 𝑐𝑜𝑠𝑥 B) 𝑦 = 𝑐𝑜𝑠2𝑥 C) 𝑦 = 𝑠𝑖𝑛2𝑥 D) 𝑦 = 2𝑠𝑖𝑛𝑥 𝑐𝑜𝑠𝑥
2. 𝑦 = 1 + 𝑐𝑜𝑠𝑥 is the solution of
A) y' ' y 1 B) y' ' xy x C) 2 y' ' y x D) y' ' y 1
3. The general solution of y' ' y 0 is
2
A) y A cos wx B sin wx B) y ( Ax B) cos wx C) y ( A Bx) sin wx D) y Ae wx Be wx
4. The frequency of the harmonic oscillation is
2
A) f 0 B) f 0 C) f D) f
2 0 0
5. The auxiliary equation of x 2 y' 'axy 'by 0 is
A) m2 (a 1)m b 0 B) m 2 (a 1)m b 0 C) m 2 (a 1)m b 0 D) m 2 (1 a)m b 0
6. Linearity principle does not hold for _________ and _________.
7. For the homogeneous linear ODE, the backbone of the structure is ______________
8. A function y=h(x) is a solution of a second-order linear ODE on some open interval I if h is defined and
twice differentiable throughout that interval and is such that the ODE becomes an ____________ if we
replace the unknown y by h, the derivative y ' by h' , and the second derivative y ' ' by h' ' .
Part- B (2 marks)
1. Verify that the given function are linearly independent and form a basis of solutions of the given ODE
𝑦 ′′ + 2𝑦 ′ + 𝑦 = 0, 𝑦 0 = 2, 𝑦 ′ 0 = −1, 𝑒 −𝑥 , 𝑥𝑒 −𝑥 .
2. Solve the equation 𝑦 ′′ + 36𝑦 = 0.
3. Find a general solution of 𝑦 ′′ + 4𝑦 ′ + (𝜋 2 + 4)𝑦 = 0.
4. Find a general solution of 10𝑦 ′′ − 32𝑦 ′ + 25.6𝑦 = 0.
5. Find an ODE for the basis 𝑒 2.6𝑥 , 𝑒 −4.3𝑥 .
6. Find an ODE for the basis 𝑒 −√2𝑥 , 𝑥𝑒 −√2𝑥 .
7. Find an ODE for the basis x cos(2 log x), x sin(2 log x) .
8. Find the general solution of 𝑥 2 𝑦 ′′ + 4𝑥𝑦 ′ = 0.
9. Find the general solution of 𝑥 2 𝐷2 − 3𝑥𝐷 + 4𝐼 𝑦 = 0.
10. Find the Wronskian of the functions 𝑥 3 , 𝑥 2 .
11. Find the Wronskian of the functions 𝑒 −𝑥 𝑐𝑜𝑠𝜔𝑥, 𝑒 −𝑥 𝑠𝑖𝑛𝜔𝑥.
Part- C (Descriptive)
1. Solve the IVP y' ' y'6 y 0, y(0) 10, y' (0) 0 . Check that your answer satisfies the ODE as well as the
initial conditions.
2. Solve the IVP 9 y' '30 y'25 y 0, y(0) 3.3, y' (0) 10 .
3. Solve x 2 y' '3xy ' y 0, y(1) 3.6, y' (1) 0.4 .
4. Find the steady-state current in the RLC-circuit for the given data R 4, L 0.1H , C 0.05F , E 110V
5. Solve the given non-homogeneous linear ODE by variation of parameters x y' '2 xy '2 y x sin x .
2 3
6. Solve the given non-homogeneous linear ODE by variation of parameters
x 2 y' ' xy ' y 16 x 3 , y(1) 1, y' (1) 1 .
7. Solve the initial value problem 𝑦 ′′ + 𝑦 ′ − 6𝑦 = 0, 𝑦 0 = 10, 𝑦 ′ 0 = 0.
8. Solve the initial value problem 𝑦 ′′ + 𝑦 ′ − 6𝑦 = 0, 𝑦 0 = 10, 𝑦 ′ 0 = 0.
9. Solve the initial value problem 𝑥 2 𝐷2 − 3𝑥𝐷 + 4𝐼 𝑦 = 0, 𝑦 1 = −𝜋, 𝑦 ′ 1 = 2𝜋.
10. Solve the initial value problem 𝑥 2 𝑦 ′′ + 3𝑥𝑦 ′ + 0.75𝑦 = 0, 𝑦 1 = 1, 𝑦 ′ 1 = −2.5.
11. Solve the non-homogeneous linear ODE by variation of parameters 𝑥 2 𝑦 ′′ − 𝑥𝑦 ′ − 3𝑦 = 𝑥 2 .
12. Solve the non-homogeneous linear ODE by variation of parameters 𝑦 ′′ − 4𝑦 ′ + 5𝑦 = cosec 𝑥.
2 2 −𝑥
13. Solve the non-homogeneous linear ODE by variation of parameters (𝐷 − 2𝐷 + 𝐼)𝑦 = 6𝑥 𝑒 .
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