Let y 1 x and y 2 x are the solutions of the differential equation y cos x - e x cos x d x = d y - e x d x sin x and y 1 x passes through the point π 2 , e π 2 and y 2 x passes through the point - π 2 , 1 , then which of the following is/are true?

Area between the curves y 1 x , y 2 x and the lines x = 0 and x = π is 2 1 - e - π 2 square units

The equation y 1 x = y 2 x has exactly 2 solutions is 0 , 2 π

Area between the curves y 1 x , y 2 x & the lines x = 0 and x = π is e - π 2 - 1 square units

Let y 1 x and y 2 x are the solutions of the differential equation y cos x - e x cos x d x = d y - e x d x sin x and y 1 x passes through the point π 2 , e π 2 and y 2 x passes through the point - π 2 , 1 , then which of the following is/are true?

Area between the curves y 1 x , y 2 x and the lines x = 0 and x = π is 2 1 - e - π 2 square units

A solution curve of differential equation x 2 + x y + 4 x + 2 y + 4 d y d x - y 2 = 0 , x > 0 passes through the point ( 1 , 3 ) . Then the solution curve -

intersects y = x + 2 exactly at two points

The solution of the differential equation d y d x - y + 3 x log e ( y + 3 x ) + 3 = 0 is (where C is a constant of integration)

x - 1 2 log e ( y + 3 x ) 2 = C

HARD

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Let $y_{1} (x)$ and $y_{2} (x)$ are the solutions of the differential equation $(y \cos x - e^{x} \cos x) d x = (d y - e^{x} d x) \sin x$ and $y_{1} (x)$ passes through the point $(\frac{π}{2}, e^{\frac{π}{2}})$ and $y_{2} (x)$ passes through the point $(\frac{- π}{2}, 1)$ , then which of the following is/are true?

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Important Questions on Differential Equation

HARD

Mathematics>Integral Calculus>Differential Equation>Exact Differential Equations

Let

y = y (x)

be the solution of the differential equation

(x - x^{3}) d y = (y + y x^{2} - 3 x^{4}) d x, x > 2

y (3) = 3,

then

y (4)

is equal to:

HARD

Mathematics>Integral Calculus>Differential Equation>Exact Differential Equations

The solution of the differential equation

y d x - (x + 2 y^{2}) d y = 0

x = f (y)

. If

f (- 1) = 1,

then

f (1)

is equal to

MEDIUM

Mathematics>Integral Calculus>Differential Equation>Exact Differential Equations

x d y = y (d x + y d y), y (1) = 1

and

y (x) > 0

, then the value of

y (- 3)

HARD

Mathematics>Integral Calculus>Differential Equation>Exact Differential Equations

For

c

is the arbitrary constant, the solution of the differential equation

(x^{2} - y x^{2}) \frac{d y}{d x} + (y^{2} + x y^{2}) = 0

HARD

Mathematics>Integral Calculus>Differential Equation>Exact Differential Equations

The solution of the differential equation

y d x - x d y = x y d x

is ….

HARD

Mathematics>Integral Calculus>Differential Equation>Exact Differential Equations

Let

y = y (x)

be the solution of the differential equation

x d y = (y + x^{3} \cos x) d x

with

y (π) = 0,

then

y (\frac{π}{2})

is equal to:

HARD

Mathematics>Integral Calculus>Differential Equation>Exact Differential Equations

y = f (x)

, passes through the point

(1, - 1)

and satisfies the equation

y (1 + x y) d x = x d y,

then

f (\frac{1}{2})

is equal to

HARD

Mathematics>Integral Calculus>Differential Equation>Exact Differential Equations

A solution curve of differential equation

(x^{2} + x y + 4 x + 2 y + 4) \frac{d y}{d x} - y^{2} = 0, x > 0

passes through the point

(1, 3)

. Then the solution curve -

HARD

Mathematics>Integral Calculus>Differential Equation>Exact Differential Equations

The solution of the differential equation

\frac{d y}{d x} - \frac{y + 3 x}{\log_{e} (y + 3 x)} + 3 = 0

is
(where

C

is a constant of integration)

EASY

Mathematics>Integral Calculus>Differential Equation>Exact Differential Equations

The solution of the differential equation

y d x - x d y + 3 x^{2} y^{2} e^{x^{3}} d x = 0

satisfying

y = 1

when

x = 1

, is

HARD

Mathematics>Integral Calculus>Differential Equation>Exact Differential Equations

Let

y = y (x)

be the solution of the differential equation

(x + 1) y^{'} - y = e^{3 x} {(x + 1)}^{2}

, with

y (0) = \frac{1}{3}

. Then, the point

x = - \frac{4}{3}

for the curve

y = y (x)

HARD

Mathematics>Integral Calculus>Differential Equation>Exact Differential Equations

A curve amongst the family of curves represented by the differential equation,

(x^{2} - y^{2}) d x + 2 x y d y = 0

which passes through

(1, 1)

, is

EASY

Mathematics>Integral Calculus>Differential Equation>Exact Differential Equations

The general solution of the differential equation

(x^{2} y^{2} + y) d x - (x - 2 x^{3} y) d y = 0

HARD

Mathematics>Integral Calculus>Differential Equation>Exact Differential Equations

Let

y = y (x)

be the solution of the differential equation

\frac{d y}{d x} = 1 + x e^{y - x}, - \sqrt{2} < x < \sqrt{2}, y (0) = 0

, then the minimum value of

y (x), x \in (- \sqrt{2}, \sqrt{2})

is equal to :

MEDIUM

Mathematics>Integral Calculus>Differential Equation>Exact Differential Equations

The differential equation whose linearly independent solutions are

c o s 2 x, s i n 2 x, e^{- x},

HARD

Mathematics>Integral Calculus>Differential Equation>Exact Differential Equations

Let

f : (0, π) \to R

be a twice differentiable function such that

\lim_{t \to x} \{\frac{f (x) \sin t - f (t) \sin x}{t - x}\} = \sin^{2} x

for all

x \in (0, π)

. If

f (\frac{π}{6}) = - \frac{π}{12}

, then which of the following statement(s) is (are) TRUE ?

HARD

Mathematics>Integral Calculus>Differential Equation>Exact Differential Equations

Let

f : R \to R a n d g : R \to R

be two non-constant differentiable functions. If

f^{'} (x) = (e^{(f (x)) - g (x)}) g^{'} (x) f o r a l l x \in R

, and

f (1) = g (2) = 1

then which of the following statement(s) is (are) TRUE?

HARD

Mathematics>Integral Calculus>Differential Equation>Exact Differential Equations

Let

f

be a twice differentiable function defined on

R

such that

f (0) = 1, f^{'} (0) = 2

and

f^{'} (x) \neq 0

for all

x \in R .

|\begin{array}{l} f (x) & f^{'} (x) \\ f^{'} (x) & f^{''} (x) \end{array}| = 0,

for all

x \in R,

then the value of

f (1)

lies in the interval

HARD

Mathematics>Integral Calculus>Differential Equation>Exact Differential Equations

If a curve

y = f (x)

passes through the point

(1, - 1)

and satisfies the differential equation,

y (1 + x y) d x = x d y,

then

f (- \frac{1}{2})

is equal to

EASY

Mathematics>Integral Calculus>Differential Equation>Exact Differential Equations

What is the general solution of the differential equation

y d x - (x + 2 y^{2}) d y = 0 ?

Let y1x and y2x are the solutions of the differential equation ycosx-excosxdx=dy-exdxsinx and y1x passes through the point π2,eπ2 and y2x passes through the point -π2,1, then which of the following is/are true?

Important Questions on Differential Equation

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