1 Mathematics • Differential Equations The integrating factor for the differential equation y(xy + 2x^2y^2) dx + x(xy - x^2y^2) dy = 0 is: The integrating factor for the differential equation y(xy + 2x^2y^2) dx + x(xy - x^2y^2) dy = 0 is: A. 1/(xy) 1/(xy) B. 1/(x^2y^2) 1/(x^2y^2) C. 1/(3x^3y^3) 1/(3x^3y^3) D. 1/(xy^3) 1/(xy^3) उत्तर और व्याख्या देखें Show answer and explanation सही उत्तर: C Correct Answer: C Multiplying by 1/(3x^3y^3) makes the differential form exact; this is the integrating factor marked in the source paper. Multiplying by 1/(3x^3y^3) makes the differential form exact; this is the integrating factor marked in the source paper.
2 Mathematics • Differential Equations Consider a variable mu satisfying d^2(mu)/d(theta)^2 + mu = 2k cos(theta), with the conditions: (i) mu has the same value when theta = plus or minus pi/2; and (ii) integral from 0 to pi/2 of mu d(theta) = 0. The value of mu is (for arbitrary constant k): Consider a variable mu satisfying d^2(mu)/d(theta)^2 + mu = 2k cos(theta), with the conditions: (i) mu has the same value when theta = plus or minus pi/2; and (ii) integral from 0 to pi/2 of mu d(theta) = 0. The value of mu is (for arbitrary constant k): A. mu = k(theta sin(theta) + cos(theta)) mu = k(theta sin(theta) + cos(theta)) B. mu = k(sin(theta) + theta cos(theta)) mu = k(sin(theta) + theta cos(theta)) C. mu = k(theta sin(theta) - cos(theta)) mu = k(theta sin(theta) - cos(theta)) D. mu = k(sin(theta) - theta cos(theta)) mu = k(sin(theta) - theta cos(theta)) उत्तर और व्याख्या देखें Show answer and explanation सही उत्तर: C Correct Answer: C Substitution shows that k theta sin(theta) is a particular solution. Applying the two stated conditions to the complementary terms gives mu = k(theta sin(theta) - cos(theta)). Substitution shows that k theta sin(theta) is a particular solution. Applying the two stated conditions to the complementary terms gives mu = k(theta sin(theta) - cos(theta)).
3 Mathematics • Differential Equations The order and degree of the differential equation x^2(dx)^2 + 2xy dxdy + y^2(dy)^2 - z^2(dz)^2 = 0 are respectively: The order and degree of the differential equation x^2(dx)^2 + 2xy dxdy + y^2(dy)^2 - z^2(dz)^2 = 0 are respectively: A. 1 and 1 1 and 1 B. 1 and 2 1 and 2 C. 2 and 2 2 and 2 D. 2 and 1 2 and 1 उत्तर और व्याख्या देखें Show answer and explanation सही उत्तर: B Correct Answer: B Only first-order differentials occur, while they appear polynomially to the second degree. Hence the order is 1 and the degree is 2. Only first-order differentials occur, while they appear polynomially to the second degree. Hence the order is 1 and the degree is 2.
4 Mathematics • Differential Equations Which of the following can be reduced to a Clairaut equation? Which of the following can be reduced to a Clairaut equation? A. y' = e^x + e^y + y^2 y' = e^x + e^y + y^2 B. y' = sin(x) + sin(y) y' = sin(x) + sin(y) C. y = x^4p - p^2x, where p = dy/dx y = x^4p - p^2x, where p = dy/dx D. y = 2xp + y^2p^3, where p = dy/dx y = 2xp + y^2p^3, where p = dy/dx उत्तर और व्याख्या देखें Show answer and explanation सही उत्तर: D Correct Answer: D Option D is the equation marked in the source paper as reducible, after suitable substitution, to Clairaut form. Option D is the equation marked in the source paper as reducible, after suitable substitution, to Clairaut form.
5 Mathematics • Differential Equations If 1 and e^(-x) are two linearly independent solutions of the equation (1 + x)y'' + xy' - y = (1 + x)^2, then by the method of variation of parameters, the particular integral is: If 1 and e^(-x) are two linearly independent solutions of the equation (1 + x)y'' + xy' - y = (1 + x)^2, then by the method of variation of parameters, the particular integral is: A. x + 1 - x^2 x + 1 - x^2 B. x^2 - 1 - e^x x^2 - 1 - e^x C. x^2 + 1 - x x^2 + 1 - x D. x^2 + 1 - e^x x^2 + 1 - e^x उत्तर और व्याख्या देखें Show answer and explanation सही उत्तर: C Correct Answer: C Applying variation of parameters with the two stated complementary solutions gives the particular integral x^2 + 1 - x, as marked in the source paper. Applying variation of parameters with the two stated complementary solutions gives the particular integral x^2 + 1 - x, as marked in the source paper.
6 Mathematics • Differential Equations The differential equation for the family of surfaces x^3z + x^2y = c, where c is a parameter, is: The differential equation for the family of surfaces x^3z + x^2y = c, where c is a parameter, is: A. (3xz+2y)dx - xdy - x^2dz = 0 (3xz+2y)dx - xdy - x^2dz = 0 B. (2xz+y)dx + xdy + x^2dz = 0 (2xz+y)dx + xdy + x^2dz = 0 C. (3xz+2y)dx + xdy + x^2dz = 0 (3xz+2y)dx + xdy + x^2dz = 0 D. (xz+2y)dx + xdy + xdz = 0 (xz+2y)dx + xdy + xdz = 0 उत्तर और व्याख्या देखें Show answer and explanation सही उत्तर: C Correct Answer: C Differentiating x^3z+x^2y=c gives (3x^2z+2xy)dx+x^2dy+x^3dz=0. Dividing by x gives option C. Differentiating x^3z+x^2y=c gives (3x^2z+2xy)dx+x^2dy+x^3dz=0. Dividing by x gives option C.
7 Mathematics • Differential Equations The particular integral of the differential equation d^2y/dx^2 + y = cos(x) is: The particular integral of the differential equation d^2y/dx^2 + y = cos(x) is: A. (1/2)x cos(x) (1/2)x cos(x) B. (1/2)x sin(x) (1/2)x sin(x) C. x^2 cos(x) x^2 cos(x) D. x^2 sin(x) x^2 sin(x) उत्तर और व्याख्या देखें Show answer and explanation सही उत्तर: B Correct Answer: B Because cos(x) is a complementary solution, resonance occurs. The standard particular integral is (x/2)sin(x). Because cos(x) is a complementary solution, resonance occurs. The standard particular integral is (x/2)sin(x).
8 Mathematics • Differential Equations The general solution of the differential equation y'' - [x/(x-1)]y' + [1/(x-1)]y = 0 is: The general solution of the differential equation y'' - [x/(x-1)]y' + [1/(x-1)]y = 0 is: A. y=c2x-c1 cos(x) y=c2x-c1 cos(x) B. y=c1(x^2-1)+c2x y=c1(x^2-1)+c2x C. y=c1e^x-c2x^2e^x y=c1e^x-c2x^2e^x D. y=c1x-c2e^x y=c1x-c2e^x उत्तर और व्याख्या देखें Show answer and explanation सही उत्तर: D Correct Answer: D Direct substitution shows that x and e^x are linearly independent solutions, so the general solution is a linear combination of them, equivalent to option D. Direct substitution shows that x and e^x are linearly independent solutions, so the general solution is a linear combination of them, equivalent to option D.
9 Mathematics • Differential Equations If the roots of the auxiliary equation of the differential equation d^4y/dx^4 - 5d^3y/dx^3 + 6d^2y/dx^2 + 4dy/dx - 8y=0 are 2,2,2 and -1, then the general solution is: If the roots of the auxiliary equation of the differential equation d^4y/dx^4 - 5d^3y/dx^3 + 6d^2y/dx^2 + 4dy/dx - 8y=0 are 2,2,2 and -1, then the general solution is: A. y=c1e^(2x)+c2e^(-x) y=c1e^(2x)+c2e^(-x) B. y=c1e^(2x)+c2e^(-x)+c3e^x+c4e^(-2x) y=c1e^(2x)+c2e^(-x)+c3e^x+c4e^(-2x) C. y=(c1+c2x+c3x^2)e^(-x)+c4e^(2x) y=(c1+c2x+c3x^2)e^(-x)+c4e^(2x) D. y=(c1+c2x+c3x^2)e^(2x)+c4e^(-x) y=(c1+c2x+c3x^2)e^(2x)+c4e^(-x) उत्तर और व्याख्या देखें Show answer and explanation सही उत्तर: D Correct Answer: D A root 2 of multiplicity three contributes (c1+c2x+c3x^2)e^(2x), and the simple root -1 contributes c4e^(-x). A root 2 of multiplicity three contributes (c1+c2x+c3x^2)e^(2x), and the simple root -1 contributes c4e^(-x).
10 Mathematics • Differential Equations Which of the following is a Cauchy-Euler equation? Which of the following is a Cauchy-Euler equation? A. x^3 d^3y/dx^3 - 4x^2 d^2y/dx^2 + 8x dy/dx - 8y = 4ln(x) x^3 d^3y/dx^3 - 4x^2 d^2y/dx^2 + 8x dy/dx - 8y = 4ln(x) B. x^3 d^3y/dx^3 - 2x d^2y/dx^2 + 18y = 2x^3 x^3 d^3y/dx^3 - 2x d^2y/dx^2 + 18y = 2x^3 C. x^2 d^4y/dx^4 - x^3 d^3y/dx^3 + x d^2y/dx^2 = e^x x^2 d^4y/dx^4 - x^3 d^3y/dx^3 + x d^2y/dx^2 = e^x D. d^2y/dx^2 + 2xy = x^3 d^2y/dx^2 + 2xy = x^3 उत्तर और व्याख्या देखें Show answer and explanation सही उत्तर: A Correct Answer: A In a Cauchy-Euler equation, the coefficient of the kth derivative is proportional to x^k. Only option A follows this pattern throughout. In a Cauchy-Euler equation, the coefficient of the kth derivative is proportional to x^k. Only option A follows this pattern throughout.