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Let ( y  C)^{2} = Cx be the primitive of The no. of integral curves which will pass through (1, 2) is,
Which of the following transformations reduce the differential equation into the form
Given differential equation is
Here which shows that the transformation t= 1/log z
reduce the given differential equation into the form
then its solution is
Now,
The solution of the differential equation ^{}
where y(0) = 0 and y'(0) = 2 is
Auxiliary equation is m^{2}  m2 = 0
m=1,2
∴ C.F = c_{1}e^{x }+c_{2}e^{2x}
Solving these, we get c_{1}= 1, c_{2}= 1
∴ the required solution is y = e^{x}  e^{2x}+ xe^{2x}
Correct Answer : d
Explanation :
The differential equation representing the family of circles touching xaxis at the origin is
The equation of the family of the circles touching xaxis at the origin is
x^{2} + (ya)^{2} = a^{2}
A curve passing through (2,3) and satisfying the differential equation
differential w.r. to x
The differential equation
(3a^{2}x^{2} + by cos x)dx + (2 sin x  4ay^{3})dy = 0 is exact for
Given differential equation is
(3a^{2}x^{2} + by cos x)dx + (2 sin x  4ay^{3})dy = 0
since it si exact if
Here M = 3a^{2}x^{2} + bycosx
N = 2sinx  4ay^{3}
then bcosx = 2cosx
=> cosx(b  2) = 0
=> b= 2[∵ cos x ≠ 0]
which shows that given differential equation is exact for any value of a but b = 2
The integrating factor of the equation (1 + y^{2})dx = (tan^{1}y  x)dy is 
The given equation can be rewritten as
Integrating
Putting y^{2} = v so that 2y (dy/dx) = (dv/dx). We then get
which is linear. Its I.F =
x=0, y=1/2 (1) ⇒ 2 = 3 +c ⇒ c=1
The slope of a curve at any point is the reciprocal of twice the ordinate at the point and it passes through the point (4, 3). The equation of the curve is
We have,
Slope = dy/dx ⇒ dy/dx = 1/2y ⇒ 2 y dy = dx
Integrating both sides, we get y^{2} = x + C
This passes through (4, 3)
∴ 9 = 4 + C ⇒ C = 5
So, the equation of the curve is y^{2} = x + 5
Hence the given differential equation
It is clear by the definition of linear differential equation and order of differential equation, that is linear and of second order.
If (c_{1} logx + c_{2})x^{2} is the general solution of the differential equation then k equals
Given differential equation is
(1)
and it’s general solution is
(c_{1} log x + c_{2}) x^{2} ..(2)
Since (1) is homogeneous differential equation so put then (1) reduce to
It’s A.E is m^{2} + m(k  1) + 4 = 0 • • (3)
Now from (2) we have
By observation we can say that (3) has solution m = 2,
... (4)
On comparing (3) and (4). we have
k1 = 4
k=3
Consider a function g which has derivative g’(x) for every real x and which satisfies g’(0) = 2 and g(x + y) = e^{y} g(x) + e^{x} g(y) for all x and y. Then which of the followings is/are correct ?
Consider the differential equation (3x^{2}y^{4} + 2xy)dx+(2x^{3}y^{3}  x^{2})dy=0 then which of the follwing(s) is/are not an I.F
Given differential equation is
If y_{1} and y_{2} are two solution of the differential equation . Then which of the followings can be given as general soln of this differential eqn.
Option (a), (b) are clearly correct
Now we have
Now from (4) and (5)
For the given differential equation, which of the following(s) statement are true,
Given equation ca be written as,
not homogenous differential equation
compare with
Here at x = 0, P and Q both are not defined.
⇒ x=0 is a singular point of given differential equation.
Now consider x P(x) = 1/2
and both are defined and differential at x = 0
⇒ x= 0 is a regular singular point of given equation.
Given that φ(x) = x^{2} is a solution of the differential equation, then which of the following(s) is/are not its 2^{nd } L.l. solution.
we have x^{2} y"  2y = 0
If the integrating factor of the differential equation, (x^{7}y^{2 }+ 3y)dx + 3x^{8}y  x) dy = 0 is of the form x^{m}y^{n} , then the sum of value of m and n is __________.
Given differential equation is
(x^{7}y^{2 }+ 3y)dx + 3x^{8}y  x) dy = 0 ..(1)
If Integrating factor is x^{m}y^{n} then multiplying the given differential equation by I.F. , equation (1) becomes exact
i.e
is exact differential equation.
and for exact
on solving (1) and (2) we get
m=7, n=1
m+n = 7+1 = 6
The sum of order and degree of the differential equation representing the family o f parabolas whose center is (a.O) i s _______ .
The equation of the families of parabolas whose centre (a,0) is
y^{2} = 4a(x+a) ..(1)
Differentiating both sides with respect to x, we get
Dividing (i) by (ii) we get
Substituting this value of a in (2), a is eliminated and we get
which shows that order of this differential equation is one and degree is 2. The sum of order of this differential equation and degree is 3.
The value o f y as t > ∞ , for an initial value o f y(1) = 0, for the differential equation
Hence solution of equation (i) is,
But, y(1)=0, therefore
From equation (ii), we have
From equation (ii), we have
If y = f(x) be a particular solution of differential equation y" + 4y = 4 cosec^{2} 2x , then f(x) at is _________.
Consider the differential equation be its general solution and u(x) = x^{3} then the value of v(x) at x=(1) is, _________
where
v(x)= 1/6x^{2}
∴ v(x)= 1/6(1)^{2} = 0.1666666
No. of regular Singular points of the differential equation, x^{2}( x  2 )^{2} y" + 2x (x  2) y’+ (x + 1)y = 0. is __________.
In the normalized form (6,3), this is
Clearly the singular points of the differential equation are x = 0 and x = 2. We investigate them one at a time.
Consider x = 0 first. and form the functions defined by the products
of the form. The product function defined by x^{2}P_{2}(x)f is analytical at x = 0, but that defined by xP_{1}(x) is not. Thus x = 0 is an irregular singular point of Now consider x = 2. Forming the products for this point, we have
Both of the product functions thus defined are analytic at x = 2. and hence x = 2 is a regular singular point.
The value of I.F of the differential equation, at x = 2 is, _______.
The directional derivative of f(x,y) = x^{2} + xy at P_{o}(1,2) in the direction of the unit vector
The derivative of f at P_{0}(x_{0},y_{o}) in the direction of the unit vector u = u_{1i} + u_{2}j is the number
Find the directional Derivative of f(x,y) = x^{2} sin 2y at the point
Find the derivation of f(x,y,z) = x^{3}  xy^{2} z at P_{o} (1,1,0) in the direction of v= 2i  3j+ 6k
the partial derivatives of f at P_{0} are
The gradient of f ata p_{0} is
The derivative of f at P_{0 }in the direction of v is therefore
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