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Right ANSWER is (d) XY = (x – 1)(x^3/3 + C)

The best EXPLANATION: x(x – 1)dy/dx – y = x^2(x – 1)^2

or, dy/dx – 1/(x(x – 1))*y = x(x – 1) ……(1)

we have, ∫- 1/(x(x – 1)) dx = – ∫ [1/(x – 1) – 1/x] dx

= – [log(x – 1) – log x]

= log x – log(x – 1)

= log(x/(x – 1))

Thus, integrating factor = e^∫- 1/(x(x – 1)) dx = x/(x – 1)

Thus, multiplying both SIDES of (1) by x/(x – 1), we get,

x/(x – 1)*dy/dx – 1/(x – 1)^2 * y = x^2

or, d/dx[x/(x – 1) * y] = x^2 …….(2)

integrating both sides of (2) we get,

x/(x – 1) * y = ∫x^2dx = x^3/3 + c

or xy = (x – 1)(x^3/3 + c)

The correct ANSWER is (a) 3x – y + LOG|2x + 3y – 3| = -c/3

Easy EXPLANATION: dy/dx = (6x + 9Y – 7)/(2x + 3y – 6)

So, dy/dx = (3(2x + 3y) – 7)/(2x + 3x – 6)……….(1)

Now, we put, 2x + 3y = z

Therefore, 2 + 3dy/dx = dz/dx [differentiating with respect to x]

Or, dy/dx = 1/3(dz/dx – 2)

Therefore, from (1) we get,

1/3(dz/dx – 2) = (3z – 7)/(z – 6)

Or, dz/dx = 2 + (3(3z – 7))/(z – 6)

= 11(z – 3)/(z – 6)

Or, (z – 6)/(z – 3) dz = 11 dx

Or, ∫(z – 6)/(z – 3) dz = ∫11 dx

Or, ∫(1 – 3/(z – 3)) dz = 11x + c

Or, z – log |z – 3| = 11x + c

Or, 2x + 3y – 11x – 3log|2x + 3y -3| = c

Or, 3y – 9x – 3log|2x + 3y – 3| = c

Or, 3x – y + log|2x + 3y – 3| = -c/3

Correct answer is (b) 2

To explain I would say: Centre of the circles can be taken as (a, a) and radius as r for some REAL numbers a and r.

Thus, the FAMILY is a two parameter family.

Hence, order of the corresponding differential EQUATION is 2.

Correct choice is (a) 10x^3+12x-3y^2+C=0

To explain I would SAY: Given that, \(\FRAC{dy}{dx}=5x^2+2\)

Separating the variables, we get

dy=(5x^2+2)dx –(1)

Integrating both sides of (1), we get

\(\INT y \,dy=\int 5x^2+2 \,dx\)

\(\frac{y^2}{2}=\frac{5x^3}{3}+2x+C_1\)

3y^2=\(10x^3+12x+6C_1\)

10x^3+12x-3y^2+C=0 (where 6C1=C)

Right ANSWER is (b) 0

Best explanation: The number of arbitrary constants for a PARTICULAR solution of n^th ORDER DIFFERENTIAL equation is ALWAYS zero.

Correct answer is (d) y=\(\frac{(2x-1)^2}{3}\)

EXPLANATION: Given that, \(\frac{dy}{dx}+8x=16x^2+4\)

\(\frac{dy}{dx}=16x^2-8x+4\)

\(\frac{dy}{dx}=(4x-2)^2\)

Separating the VARIABLES, we get

dy=(4x-2)^2 dx

Integrating both sides, we get

\(\INT dy=\int (4x-2)^2 \,dx\)

y=\(\frac{(4x-2)^2}{12}+C\)

y=\(\frac{(2x-1)^2}{3}+C\) –(1)

Given that, y=1/3 when x=1

Therefore, EQUATION (1) becomes,

\(\frac{1}{3}=\frac{(2(1)-1)^2}{3}+C\)

\(C=\frac{1}{3}-\frac{1}{3}\)=0

Hence, the PARTICULAR solution for the given differential solution is y=\(\frac{(2x-1)^2}{3}\).

Correct CHOICE is (d) 3 cos⁡X+2 sin⁡y=C

The best I can explain: Given that, \(\frac{dy}{dx}=\frac{3 \,sec⁡\,y}{2cosec \,x}\)

\(\frac{2 \,dy}{sec⁡ \,y}=\frac{3dx}{COSEC\,⁡x}\)

Separating the variables, we get

2 cos⁡y dy=3 sin⁡x dx

Integrating both sides, we get

∫ 2 cos⁡y dy = ∫ 3 sin⁡x dx

2 sin⁡y=3(-cos⁡x)+C

3 cos⁡x+2 sin⁡y=C

Correct choice is (c) Order -1, Degree-1

The best I can EXPLAIN: In the given D.E, the highest order DERIVATIVE is y’. Therefore, the order is 1. The given D.E 7y’-3y=0 is a polynomial equation in y’ and the power raised to the highest derivative is 1. HENCE, the degree is 1.

The CORRECT choice is (a) True

Easy explanation: The given statement is true.

Consider the function y=8 sin⁡2x

Differentiating w.r.t x, we get

\(\FRAC{dy}{dx}\)=16 cos⁡2x –(1)

Differentiating (1) w.r.t x, we get

\(\frac{d^2 y}{dx^2}\)=-32 sin⁡2x

\(\frac{d^2 y}{dx^2}\)=-4(8 sin⁡2x )=-4y

⇒\(\frac{d^2 y}{dx^2}\)+4y=0.

Correct answer is (c) -4

For explanation I would say: (D + 1)^2y = 0

Or, (D^2 + 2D+ 1)y = 0

=> d^2y/dx^2 + 2dy/dx + y = 0……….(1)

Let y = e^mx be a trial solution of equation (1). Then,

=> dy/dx = me^mx and d^2y/dx^2 = m^2e^mx

Clearly, y = e^mx will SATISFY equation (1). Hence, we have

=> m^2.e^mx + 2M.e^mx + e^mx = 0

Or, m^2 + 2m + 1 = 0 (as, e^mx ≠ 0)………..(2)

Or, (m + 1)^2 = 0

=> m = -1, -1

So, the ROOTS of the AUXILIARY equation (2) are real and equal. Therefore, the general solution of equation (1) is

y = (A + Bx)e^-x where A and B are two independent arbitrary constants ……….(3)

Given, y = 2 loge 2 when x = loge 2

Therefore, from (3) we get,

2 loge 2 = (A + B loge2)e^-x

Or, 1/2(A + B loge2) = 2 log e2

Or, A + B loge2 = 4 loge2……….(4)

Again y = (4/3) loge3 when x = loge3

So, from (3) we get,

4/3 loge3 = (A + Bloge3)

Or, A + Bloge3 = 4loge3……….(5)

Now, (5) – (4) gives,

B(loge3 – loge2) = 4(loge3 – loge2)

=> B = 4

Putting B = 4 in (4) we get, A = 0

Thus the required solution of (1) is y = 4xe^-x

So, C = 4

Right CHOICE is (B) \(\frac{d^2 y}{dx^2}+(\frac{dy}{dx})^2\)=0

The best I can explain: Consider the function y=log⁡X

DIFFERENTIATING w.r.t x, we get

\(\frac{dy}{dx}=\frac{1}{x} \)–(1)

Differentiating (1) w.r.t x, we get

\(\frac{d^2 y}{dx^2}=-\frac{1}{x^2} \)

∴\(\frac{d^2 y}{dx^2}+(\frac{dy}{dx})^2=-\frac{1}{x^2}+(\frac{1}{x})^2\)

=-\(\frac{1}{x^2}+\frac{1}{x^2}\)=0.

The correct option is (d) x + y = 2

The best explanation: The EQUATION of TANGENT to the curve y = f(x), at point (x, y), is

Y – y = dy/dx * (X – x) …..(1)

Where it meets the x axis, Y = 0 and X = (x – y/(dy/dx))

Where it meets the y axis, X = 0 and Y = (y – x/(dy/dx))

Also, the area of the triangle formed by (1) with the coordinate axes is 2, so that,

(x – y/(dy/dx))* (y – x/(dy/dx)) = 4

Or, (y – x/(dy/dx))^2 – 4dy/dx = 0

Or, x^2(dy/dx)^2 – 2(XY – 2)dy/dx + y^2 = 0

Solving for dy/dx we get,

dy/dx = [(xy – 2) ± √(1 – xy)]/ x^2

Let, 1 – xy = t^2

=> x(dy/dx) + y = -2t(dt/dx)

=> x^2(dy/dx) = t^2 – 1 – 2tx(dt/dx), so that (3) gives

t(x(dt/dx) – (t ± 1)) = 0

Hence, either t = 0

=>xy = 1 which is satisfied by (1, 1)

Or, x dt/dx = t ± 1

=> dx/x = dt/t ± 1

=> t ± 1 = cx

For x = 1, y = 1 and t = 0

=> c = ± 1, so the solution is

t = ± (x – 1) => t^2 = (x – 1)^2

Or, 1 – xy = x^2 – 2x + 1

Or, x + y = 2

Thus, the two curves that SATISFIES are xy = 1 and x + y = 2

Correct OPTION is (a) y=7x^2

Easiest explanation: Consider the function y=7x^2

Differentiating w.r.t X, we GET

\(\FRAC{dy}{dx}\)=14x

∴\(\frac{dy}{dx}\)-14x=0

Hence, the function y=7x^2 is a SOLUTION for the differential equation \(\frac{dy}{dx}\)-14x=0

The correct choice is (d) 2

The BEST EXPLANATION: The HIGHEST order derivative in the given differential equation is \(\frac{d^2 y}{dx^2}\). THEREFORE, the order of the differential equation is 2.

Correct CHOICE is (d) 2

The best EXPLANATION: The highest derivative of the given D.E is \(\FRAC{d^2 y}{dx^2}\). THEREFORE, the order is 2.