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12th Maths Weekly Test -2 (Ordinary Differential Equations)

St. Britto Hr. Sec. School - Madurai

12th Maths Weekly Test -2 (Ordinary Differential Equations)-Aug 2020

Time : 1.30 Hours

Part - A

Multiple Choice Question - Answer All Question

The differential equation of the family of parabolas y² =4ax is

2y = x(\(\frac{dy}{dx}\))
y = 2x(\(\frac{dy}{dx}\))
y = 2x²(\(\frac{dy}{dx}\))
y² = 2x(\(\frac{dy}{dx}\))

The solution of (x² -ay)dx=(ax-y² )dy is

y=x² +y² -a(x+y)
x² - y² + x - ay = 0
x³ +y² =3ayx+c
(x² -ay)(ax-y² )=0

The I.F. of cosec x\(\frac{dy}{dx}\) ysec² x=0 is

e^{sec x}
e^{tan x}
e^{sec x tan x}
e^sec2 x

The general solution of 4 \(\frac{d^2y}{dx^2}\)+y=0 is

y = e^{\(\frac{\pi}{2}\)} [A cos \(\frac{x}{2}\)+B sin \(\frac{x}{2}\)]
y = e^{\(\frac{\pi}{2}\)} [A cos \(\frac{x}{2}\)-B sin \(\frac{x}{2}\)]
y = Acos \(\frac{x}{2}\)+Bsin\(\frac{x}{2}\)
t = Ae^{\(\frac{x}{2}\)}+Be^{\(\frac{-\pi}{2}\)}

The I.F of \(\frac{dy}{dx}-y\)tan x=cos x is

sec x
cos x
e^{tan x}
cot x

The differential equation corresponding to xy=c² where c is an arbitrary constant is ________.

xy"+x=0
y"=0
xy'+y=0
xy"-x=0

The population p of a certain bacteria decreases at a rate proportional to the population p. The differential equation corresponding to the above statement is __________.

\(\frac{dp}{dt}=\frac{k}{p}\)
\(\frac{dp}{dt}=kt\)
\(\frac{dp}{dt}=kp\)
\(\frac{dp}{dt}=-kp\)

The general solution of x \(\frac{dy}{dx}\)=y is _________.

y=cx
x²+y²=c
x²-y²=c
y=c^x

Using y = vx, the differential equation \(\frac{dy}{dx}=\frac{y}{x+\sqrt{xy}}\) is reduced to ________.

x(1+\(\sqrt{v}\) )dv=v\(\sqrt{v}\)dx
x(1-\(\sqrt{v}\) )dv=v\(\sqrt{v}\)dx
x(1+\(\sqrt{v}\) )dv=-v\(\sqrt{v}\)dx
v(1+\(\sqrt{v}\) )dx=-v\(\sqrt{v}\)dx=0

The differential equation associated with the family of concentric circles having their centres at the origin is _________.

\(\frac{dy}{dx}=\frac{-x}{y}\)
\(\frac{dy}{dx}=\frac{-y}{x}\)
\(\frac{dy}{dx}=\frac{x}{y}\)
\(\frac{dy}{dx}=\frac{y}{x}\)

The differential equation representing the family of curves y = Acos(x + B), where A and B are parameters, is

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

The order of the differential equation of all circles with centre at (h, k ) and radius ‘a’ is

2
3
4
1

The general solution of the differential equation\(\frac{dy}{dx}=\frac{y}{x}\) is

xy = k
y = k log x
y = kx
log y = kx

The solution of \(\frac{dy}{dx}\)+ p(x)y=0 is

y = ce ^{∫ pdx}
y = ce ^{− ∫ pdx}
x = ce ^{− ∫ pdy}
x = ce ^{∫ pdy}

The integrating factor of the differential equation \(\frac{dy}{dx}\)+P(x)y=Q(x)is x, then P(x)

x
\(\frac{x^2}{2}\)
\(\frac{1}{x}\)
\(\frac{1}{x^2}\)

If p and q are the order and degree of the differential equation y =\(\frac{dy}{dx}+x^3\left(\frac{d^2y}{dx^2}\right)+xy=cosx,\)when

p < q
p = q
p > q
p exists and q does not exist

The solution of the differential equation \(\frac{dy}{dx}=2xy\) is

\(y=Ce^{x^2}\)
y= 2x² +C
\(y=Ce^{-x^2}\) + C
y = x² +C

The solution of \(\frac{dy}{dx}\) = 2^y−xis

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

If sin x is the integrating factor of the linear differential equation \(\frac{dy}{dx}\) + Py = Q, Then P is

log sin x
cos x
tan x
cot x

The number of arbitrary constants in the particular solution of a differential equation of third order is

3
2
1
0

Part - B

Generic 5 - Answer All Question

A population grows at the rate of 2% per year. How long does it take for the population to double?

The slope of the tangent at p(x,y) on the curve is -\(\left(\frac{y+3}{x+2}\right)\) . If the curve passes through the origin, find
the equation of the curve.

Sovle \((x+2)\frac{dy}{dx}=x2+4x-9\).Also find the domain of the function.

Solve : \(e^\frac{dy}{dx}\)= x + 1, y(0) = 5

Solve : (x² + x² + x + 1)\(\frac{dy}{dx}\) =2x²+x

In a bank principal increases at the rate of 5% per year. In how many years Rs.1000 doubled itself.

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