Maths-

General

Easy

Question

The differential equation of all parabolas whose axis are parallel to y-axis is

$\frac{d^{3} x}{d x^{3}} = 0$
$\frac{d^{2} x}{d y^{2}} = C$
$\frac{d^{3} y}{d x^{3}} + \frac{d^{2} y}{d x^{2}} = 0$
$\frac{d^{2} y}{d x^{2}} + 2 \frac{d y}{d x} = C$

hint Hint:

We have to find the differential equation of a parabola. The axis of parabola is parallel to x-axis. It means the vertex doesn't lie at the origin. We will write the equation of parabola and then differentiate it.

The correct answer is: $fraction numerator d to the power of 3 end exponent x over denominator d x to the power of 3 end exponent end fraction equals 0$

The equation of parabola which is parallel to x-axis is given as
(x - h)² = 4a(y - k)
Here, (h,k) is the vertex of the parabola.
The focus of parabola is (0,a)
We will differentiate the equation of parabola.
$left parenthesis x space minus space h right parenthesis squared equals 4 a left parenthesis y space minus space k right parenthesis D i f f e r e n t i a t i n g space t h e space e q u a t i o n space w. r. t space x 2 left parenthesis x space minus space h right parenthesis left parenthesis 1 space minus space 0 right parenthesis equals space 4 a left parenthesis 1 right parenthesis fraction numerator d y over denominator d x end fraction space minus 0 space space space space space space space space space... left curly bracket h comma k space a n d space a space a r e space c o n s tan t s right parenthesis 2 left parenthesis x space minus space h right parenthesis equals 4 a fraction numerator d y over denominator d x end fraction T a k i n g space t h e space s e c o n d space d e r i v a t i v e 2 left parenthesis 1 space minus space 0 right parenthesis space equals space 4 a fraction numerator d squared y over denominator d x squared end fraction space space space space space space space space 2 space equals space fraction numerator d squared y over denominator d x squared end fraction W e space w i l l space d i f f e r e n t i a t e space t h e space g i v e n space e q u a t i o n space a g a i n space a n d space r e a r r a n g e space space space space space space space space space fraction numerator d cubed y over denominator d x cubed end fraction equals 0 space space$
This is the required differential equation of a parabola which has axis parallel to x-axis.

For such questions, we should know how to write the equation of a parabola. The axis of parabola can be parallel to x or y axis. So, we have to write the equation accordingly.

Solution of the differential equation $open parentheses fraction numerator d y over denominator d x end fraction close parentheses to the power of 2 end exponent minus fraction numerator d y over denominator d x end fraction open parentheses e to the power of x end exponent plus e to the power of negative x end exponent close parentheses plus 1 equals 0$ is given by

When we get multipliction of two brackets equal to zero, any of the bracket can be zero or both of the brackets can be zero. It is decided based on the conditions in the equation.

Solution of the differential equation $open parentheses fraction numerator d y over denominator d x end fraction close parentheses to the power of 2 end exponent minus fraction numerator d y over denominator d x end fraction open parentheses e to the power of x end exponent plus e to the power of negative x end exponent close parentheses plus 1 equals 0$ is given by

Maths-General

When we get multipliction of two brackets equal to zero, any of the bracket can be zero or both of the brackets can be zero. It is decided based on the conditions in the equation.

General

Maths-

The differential equation of all ellipse centred at the origin and major and minor axes along coordinate axes is

Maths-General

General

Maths-

Area of the region bounded by $y equals tan invisible function application x comma$ tangent drawn to the curve at $x equals fraction numerator pi over denominator 4 end fraction$ and the $x minus$ axis is

Maths-General

General

Maths-

Maximum value of $x left parenthesis 1 minus x right parenthesis to the power of 2 end exponent$ , when $0 less or equal than x less or equal than 2$ , is

Maths-General

General

Maths-

If x = $c o s to the power of negative 1 end exponent$ t, y = $square root of 1 minus t to the power of 2 end exponent end root$ , then $y subscript 2 end subscript equals$

Maths-General

General

Physics-

Two capacitors $C subscript 1 end subscript equals 2 mu F$ and $C subscript 2 end subscript equals 6 mu F$ in series, are connected in parallel to a third capacitor $C subscript 3 end subscript equals 4 mu F$ . This arrangement is then connected to a battery of e.m.f. = 2V, as shown in the figure. How much energy is lost by the battery in charging the capacitors

Physics-General

General

Physics-

The combination of capacitors with $C subscript 1 end subscript equals 3 mu F comma C subscript 2 end subscript equals 4 mu F$ and $C subscript 3 end subscript equals 2 mu F$ is charged by connecting AB to a battery. Consider the following statements
I. Energy stored in $C subscript 1 end subscript$ = Energy stored in $C subscript 2 end subscript$ + Energy stored in $C subscript 3 end subscript$
II. Charge on C₁ = Charge on C₂ + Charge on C₃
III. Potential drop across C₁ = Potential drop across C₂ = Potential drop across C₃
Which of these is/are correct

Physics-General

General

Physics-

Consider a parallel plate capacitor of $10 mu rightwards arrow over short leftwards arrow F$ (micro-farad) with air filled in the gap between the plates. Now one half of the space between the plates is filled with a dielectric of dielectric constant 4, as shown in the figure. The capacity of the capacitor changes to

Physics-General

General

Physics-

In the figure a capacitor is filled with dielectrics. The resultant capacitance is

Physics-General

General

Physics-

Equivalent capacitance between A and B is

Physics-General

General

Physics-

In the figure, three capacitors each of capacitance 6 $p F$ are connected in series. The total capacitance of the combination will be

Physics-General

General

Physics-

Two capacitors A and B are connected in series with a battery as shown in the figure. When the switch S is closed and the two capacitors get charged fully, then

Physics-General

General

Physics-

A parallel plate capacitor of area A, plate separation d and capacitance C is filled with three different dielectric materials having dielectric constants $k subscript 1 end subscript comma k subscript 2 end subscript$ and $k subscript 3 end subscript$ as shown. If a single dielectric material is to be used to have the same capacitance C in this capacitor, then its dielectric constant k is given by

Physics-General

General

Physics-

In the circuit here, the steady state voltage across capacitor C is a fraction of the battery e.m.f. The fraction is decided by

Physics-General

General

Physics-

What is the effective capacitance between A and B in the following figure

Physics-General

Have a query? Ask a Tutor!!

Can’t find what you’re looking for?

The differential equation of all parabolas whose axis are parallel to y-axis is

We have to find the differential equation of a parabola. The axis of parabola is parallel to x-axis. It means the vertex doesn't lie at the origin. We will write the equation of parabola and then differentiate it.

The correct answer is:

The equation of parabola which is parallel to x-axis is given as(x - h)2 = 4a(y - k)Here, (h,k) is the vertex of the parabola.The focus of parabola is (0,a)We will differentiate the equation of parabola.This is the required differential equation of a parabola which has axis parallel to x-axis.

Related Questions to study

Solution of the differential equation is given by

Solution of the differential equation is given by

The differential equation of all ellipse centred at the origin and major and minor axes along coordinate axes is

The differential equation of all ellipse centred at the origin and major and minor axes along coordinate axes is

Area of the region bounded by tangent drawn to the curve at and the axis is

Area of the region bounded by tangent drawn to the curve at and the axis is

Maximum value of , when , is

Maximum value of , when , is

If x = t, y = , then

If x = t, y = , then

Two capacitors and in series, are connected in parallel to a third capacitor . This arrangement is then connected to a battery of e.m.f. = 2V, as shown in the figure. How much energy is lost by the battery in charging the capacitors

Two capacitors and in series, are connected in parallel to a third capacitor . This arrangement is then connected to a battery of e.m.f. = 2V, as shown in the figure. How much energy is lost by the battery in charging the capacitors

Consider a parallel plate capacitor of (micro-farad) with air filled in the gap between the plates. Now one half of the space between the plates is filled with a dielectric of dielectric constant 4, as shown in the figure. The capacity of the capacitor changes to

Consider a parallel plate capacitor of (micro-farad) with air filled in the gap between the plates. Now one half of the space between the plates is filled with a dielectric of dielectric constant 4, as shown in the figure. The capacity of the capacitor changes to

In the figure a capacitor is filled with dielectrics. The resultant capacitance is

In the figure a capacitor is filled with dielectrics. The resultant capacitance is

Equivalent capacitance between A and B is

Equivalent capacitance between A and B is

In the figure, three capacitors each of capacitance 6 are connected in series. The total capacitance of the combination will be

In the figure, three capacitors each of capacitance 6 are connected in series. The total capacitance of the combination will be

Two capacitors A and B are connected in series with a battery as shown in the figure. When the switch S is closed and the two capacitors get charged fully, then

Two capacitors A and B are connected in series with a battery as shown in the figure. When the switch S is closed and the two capacitors get charged fully, then

In the circuit here, the steady state voltage across capacitor C is a fraction of the battery e.m.f. The fraction is decided by

In the circuit here, the steady state voltage across capacitor C is a fraction of the battery e.m.f. The fraction is decided by

What is the effective capacitance between A and B in the following figure

What is the effective capacitance between A and B in the following figure

With Turito Academy.

With Turito Foundation.

Get an Expert Advice From Turito.

Turito Academy

With Turito Academy.

Test Prep

With Turito Foundation.

Courses

Quick Links

The correct answer is: $fraction numerator d to the power of 3 end exponent x over denominator d x to the power of 3 end exponent end fraction equals 0$

Area of the region bounded by $y equals tan invisible function application x comma$ tangent drawn to the curve at $x equals fraction numerator pi over denominator 4 end fraction$ and the $x minus$ axis is

Area of the region bounded by $y equals tan invisible function application x comma$ tangent drawn to the curve at $x equals fraction numerator pi over denominator 4 end fraction$ and the $x minus$ axis is

Maximum value of $x left parenthesis 1 minus x right parenthesis to the power of 2 end exponent$ , when $0 less or equal than x less or equal than 2$ , is

Maximum value of $x left parenthesis 1 minus x right parenthesis to the power of 2 end exponent$ , when $0 less or equal than x less or equal than 2$ , is

If x = $c o s to the power of negative 1 end exponent$ t, y = $square root of 1 minus t to the power of 2 end exponent end root$ , then $y subscript 2 end subscript equals$

If x = $c o s to the power of negative 1 end exponent$ t, y = $square root of 1 minus t to the power of 2 end exponent end root$ , then $y subscript 2 end subscript equals$

In the figure, three capacitors each of capacitance 6 $p F$ are connected in series. The total capacitance of the combination will be

In the figure, three capacitors each of capacitance 6 $p F$ are connected in series. The total capacitance of the combination will be