JEE MATHEMATICS MCQ: DIFFERENTIAL EQUATIONS

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Let

f:[0,1] \to R

be such that

f(xy) = f(x).f(y)

, for all

x,y \in [0,1]

and

f(0) \ne 0

. If

y = y(x)

satisfies the differential equation,

\frac{{dy}}{{dx}} = f(x)\mathrm{ }

with

y(0) = 1

, then

y\left( {\frac{1}{4}} \right) + y\left( {\frac{3}{4}} \right)

is equal to

View answer and explanation

Correct option: B

Option B (3) is correct because by using the given functional equation and the differential equation, we find that y = x + 1, and thus y(1/4) + y(3/4) = 3.

More Questions from DIFFERENTIAL EQUATIONS

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Let

y=y(x)

be the solution of the differential equation

\left(3 y^{2}-5 x^{2}\right) y d x+2 x\left(x^{2}-y^{2}\right) d y=0

such that

y(1)=1

. Then

\left|(y(2))^{3}-12 y(2)\right|

is equal to : The degree and order of the differential equation whose solution is a parabola whose axis is

x

-axis, are If

{x^2} + {y^2} = 1

, then

\left( {y' = \frac{{dy}}{{dx}},y'' = \frac{{{d^2}y}}{{d{x^2}}}} \right)

x = A\cos 4t + B\sin 4t,

then

\frac{{{d^2}x}}{{d{t^2}}}

is equal to Let

y=y_{1}(x)

and

y=y_{2}(x)

be the solution curves of the differential equation

\frac{d y}{d x}=y+7

with initial conditions

y_{1}(0)=0

and

y_{2}(0)=1

respectively. Then the curves

y=y_{1}(x)

and

y=y_{2}(x)

intersect at Let

y=y(t)

be a solution of the differential equation

\frac{d y}{d t}+\alpha y=\gamma e^{-\beta t}

where,

\alpha>0, \beta>0

and

\gamma>0

. Then

\lim \limits_{{x \rightarrow \infty}} y(t)