Math, asked by Anonymous, 1 month ago

9. The differential equation whose solution is 2 2 Ax By + = 1 where A and B are arbitrary constants is of [2006] (a) second order and second degree (b) first order and second degree (c) first order and first degree (d) second order and first degree 10. The differential equation representing the family of curves 2 y cx c = + 2( ) , where c > 0, is a parameter, is of order and degree as follows : [2005] (a) order 1, degree 2 (b) order 1, degree 1 (c) order 1, degree 3 (d) order 2, degree 2 11. The differential equation for the family of circle 2 2 x y ay +- = 2 0, where a is an arbitrary constant is [2004] (a) 2 2 ( )2 x y y xy + =¢ (b) 2 2 2( ) x y y xy + =¢ (c) 2 2 ( )2 x y y xy - =¢ (d) 2 2 2( ) x y y xy - =¢ 12. The degree and order of the differential equation of the family of all parabolas whose axis is x - axis, are respectively. [2003] (a) 2, 3 (b) 2, 1 (c) 1, 2 (d) 3, 2. 13. The order and degree of the differential equation 2/3 3 3 13 4 dy d y dx dx æ ö ç ÷ + = è ø are [2002] (a) (1, 3 2 ) (b) (3, 1) (c) (3, 3) (d) (1, 2) TOPIC n General & Particular Solution of Differential Equation, Solution of Differential Equation by the Method of Separation of Variables, Solution of Homogeneous Differential Equations 14. The general solution of the differential equation 2 2 22 1+++ + dy x y x y xy dx = 0 is :[Sep. 06, 2020 (I)] (where C is a constant of integration) (a) 2 2 2 2 1 11 1 1 log 2 1 1 æ ö + + + ++ = ç ÷ + ç ÷ è ø + - e x y x C x (b) 2 2 2 2 1 11 1 1 log 2 1 1 æ ö + + + - + = + ç ÷ ç ÷ è ø + - e x y x C x (c) 2 2 2 2 1 11 1 1 log 2 1 1 æ ö + - + ++ = ç ÷ + ç ÷ è ø + + e x y x C x (d) 2 2 2 2 1 11 1 1 log 2 1 1 æ ö + - + - + = + ç ÷ ç ÷ è ø + + e x y x C x 15. If 2 1 æ ö = - ç ÷ è ø p y x cosec x is the solution of the differential equation, d 2 p( ) cosec d + = p y xy x x , 0 2 p < < x , then the function p(x) is equal to: [Sep. 06, 2020 (II)] (a) cot x (b) cosec x (c) sec x (d) tan x 16. If y = y (x) is the solution of the differential equation 5e d . e 0 2 d x y x y x + + = + satisfying y (0) = 1, then a value of y(loge 13) is : [Sep. 05, 2020 (I)] (a) 1 (b) – 1 (c) 0 (d) 2 17. The solution of the differential equation 3 3 0 log ( 3 ) e dy y x dx yx + - += + is : [Sep. 04, 2020 (II)] (where C is a constant of integration.) (a) 1 2 (log ( 3 )) 2 e x yx C - += (b) log ( 3 ) e x yxC - += (c) 1 2 3 (log ) 2 e yx x C + - = (d) 2log ( 3 ) e x yxC - += 18. Let f :(0, ) (0, ) ¥® ¥ be a differentiable function such that f (1) = e and 22 22 ( ) () lim 0. t x tf x xf t ® t x - = - If f (x) = 1, then x is equal to : [Sep. 04, 2020 (II)] (a) 1 e (b) 2e (c) 1 2e (d) e EBD_8344 Downloaded from @Freebooksforjeeneet

Answers

Answered by pulakmath007

SOLUTION

TO CHOOSE THE CORRECT OPTION

The differential equation whose solution is Ax² + By² = 1 where A and B are arbitrary constants is of

(a) second order and second degree

(b) first order and second degree

(d) second order and first degree

CONCEPT TO BE IMPLEMENTED

Differential Equation :

A differential equation is an equation which involves differential coefficients or differentials

Order of a differential equation :

The order of a differential equation is the order of the highest derivative appearing in it.

Degree of a differential equation :

The degree of a differential equation is the degree of the highest derivative occuring in it after the equation has been expressed in a form free from radicals and fractions as far as the derivatives are concerned

EVALUATION

Here the given equation is

$\sf{A {x}^{2} + B {y}^{2} = 1 \: \: \: \: \: \: \: \: - - - (1)}$

Differentiating both sides with respect to x we get

$\sf{2Ax + 2Byy_1 = 0 }$

$\sf{Ax + Byy_1 = 0 \: \: \: \: \: \: \: \: - - - (2)}$

Again Differentiating both sides with respect to x we get

$\sf{A + B(yy_2 + {y_1}^{2} )= 0 \: \: \: \: \: \: \: \: - - - (3)}$

$\sf{ \implies \: A = - B(yy_2 + {y_1}^{2} ) }$

Putting the value of A in Equation 2 we get

$\sf{ - Bx(yy_2 + {y_1}^{2} ) + Byy_1 = 0 }$

$\sf{ \implies \: x(yy_2 + {y_1}^{2} ) - yy_1 = 0 }$

This is the required differential equation

This equation is of order 2 and degree 1

FINAL ANSWER

Hence the correct option is

(d) second order and first degree

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