Download SGBAU BSc 2019 Summer 2nd Sem Mathematics Differential Equations Ordinary n Partial Question Paper

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B.Sc. Part-1 (Semester-II) Examination

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MATHEMATICS

(Differential Equations : Ordinary & Partial)

Paper—III

Time : Three Hours] [Maximum Marks : 60

Note :— (1) Question No. 1 is compulsory. Solve it in ONE attempt only.

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(2) Attempt ONE question from each unit.

Choose the correct alternative :

1. The roots of the equation (D² — 4D + 13)y = 0 are :
   1. distinct and real
   2. real and equal
   3. complex and repeated

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   4. None of these
2. A linear equation of first order is of the form Y' + PY = Q in which ?
   1. P is function of Y
   2. P and Q are function of X
   3. P is function of X and Q is function of Y

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   4. None of these
3. The condition for the partial differential equation f(x, y, z, p, Q) = 0 and g(x, y, z, p, Q) =0 to be compatible is that :
   1. None of these
4. The D.E. is called :
   1. Partial differential equation

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   2. Ordinary differential equation
   3. Total differential equation
   4. Linear differential equation
5. An equation of the form Pp + Qq = R where P, Q, R are the functions of X, Y, Z is called :
   1. Lagrange’s equation

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   2. Jacobi’s equation
   3. Charpit’s equation
   4. Clairaut’s equation
6. The particular solution of DE W" + PW' + QW = 0 is y = e^x iff :
   1. P+Q=0

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   2. 1+P+Q=0
   3. 1-P+Q=0
   4. 1+P+Q=0
7. The solution of PDE (D — mD')z = 0 is :
   1. z=F(y + mx)

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   2. z=F(y - mx)
   3. None of these
8. of PDE is :
   1. F(x,y,z, p) =0
   2. F(x,y,z,q) =0

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   3. F(x,y,z,p,q) =0
   4. F(y,z,p,q) =0
9. The complete integral of F(x, p) = G(y. q) is :
   1. z= ∫ h(x, a)dx
   2. ∫ k(y, a)dy

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   3. z= ∫ h(x, a)dx + ∫ k(y, a)dy +b
   4. None of these
10. The DE Mdx + Ndy = 0 is exact iff

UNIT—I

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2. (a) Show that the D.E. : (sin x sin y — x e^y)dy = (e^y + cos x - cos y)dx is exact and hence solve it.
   (b) Find the orthogonal trajectory of rⁿ = aⁿ cos nθ.
3. (p) Solve the D.E. : (1 + x³)dy + 2xy dx = cot x dx.
   (q) Solve

UNIT—II

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4. (a) Solve the D.E. (D² - 4)y = e^x
   (b) Solve the D.E. (x²D² - 3xD - 3)y = x² sin(log x).
5. (p) Solve the D.E. (x²D² — xD + 4)y = cos(log x).
   (q) Solve the D.E.

UNIT—III

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6. (a) Solve the system of D.E : D²x — 2y = 0 and D²y + 2x = 0.
   (b) Solve the D.E. by variation of parameter.
7. (p) Solve x²y" + xy' + y = sin(log x) by changing the independent variable X to z = log x.
   (q) Solve the following D.E. by removing the first derivative :

UNIT—IV

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8. (a) Solve :
   (b) Find the complete integral of z = p²x + q.
9. (p) Find the general solution of PDE x²p + y²q = (x + y)z.
   (q) Solve the PDE p² + q² = k²

UNIT—V

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10. (a) Solve the D.E. (D² + 3DD' + 2D'²)z = x +y.
    (b) Solve by Charpits method pxy + pq + qy = yz.
11. (p) The PDE z = px + qy is compatible with any equation f(x, y, z, p, q) = 0 where f is homogeneous in x, y, z. Prove this.
    (q) Find a real function v of x and y, reducing to zero when y = 0 and satisfying

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