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12th Maths Monthly Test - 3 (Ordinary Differential Equations)

St. Britto Hr. Sec. School - Madurai

12th Maths Monthly Test - 3 (Ordinary Differential Equations)-Aug 2020

Time : 2.30 Hours

Part - A

Generic 2 - Answer All Question

Show that each of the following expressions is a solution of the corresponding given differential
equation.y = ae^x + be^-x ; y − y = 0

Solve: \(\frac{dy}{dx}=\frac{2x}{x^2+1}\)
A differential equation, determine its order, degree (if exists)

\(y\left(\frac{dy}{dx}\right)=\frac{x}{\left(\frac{dy}{dx}\right)+\left(\frac{dy}{dx}\right)^3}\)

A differential equation, determine its order, degree (if exists)

\(x^2\frac{d^2y}{dx^2}+\left[1+\left(\frac{dy}{dx}\right)^2\right]^\frac{1}{2}=0\)

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

Solve the Linear differential equation:

\(\frac{dy}{dx}=\frac{sin^2x}{1+x^3}-\frac{3x^2}{1+x^{3^y}}\)

A differential equation, determine its order, degree (if exists)

\(\frac{d^2y}{dx^2}+5\frac{dy}{dx}+∫ydx = x^3\)

Form the D.E of family of parabolas having vertex at the origin and axis along positive y-axis.

A differential equation, determine its order, degree (if exists)

\(\left(\frac{d^3y}{dx^3}\right)^\frac{2}{3}-3\frac{d^2y}{dx^2}+5\frac{dy}{dx}+4=0\)

Find the differential equations of the family of all the ellipses having foci on the y-axis and centre at
the origin.
Solve: \(X\frac{dy}{dx}=x+y\)

y=cx=c² is the general solution of the differential equation.
1) y'=c
2) y=y'x-(y')²
3) y"=0
4) (y')² -xy'+y=0

Show that x²+y² =r² , where r is a constant, is a solution of the differential equation\(\frac{dy}{dx}\)=\(-\frac{x}{y}\)

A differential equation, determine its order, degree (if exists)

\(\frac{dy}{dx}+xy=cotx\)

Determine the order and degree of\(\frac{\left[1+\left(\frac{dy}{dx}\right)^2\right]}{\frac{d^2y}{dx^2}}\)=k

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

1) xdy+y dx=x dx+y dy

2) ∫ d(xy) = ∫ xdx + ∫ ydy

3) x² -y² +2xy=c
4) x² -y² -2xy=c

Find value of m so that the function y =e^mx is a solution of the given differential equation.
y '+ 2y = 0

Part - B

Generic 3 - Answer All Question

Solve:\(\frac{dy}{dx}=\frac{1-cosc}{1+cosx}\)

Solve: \(X\frac{dy}{dx}+2y=x^2\)

Show that y = a cos(log x) + bsin (log x), x > 0 is a solution of the differential equation x² y"+ xy'+y= 0.

Solve: \(\frac{dy}{dx}\)=(4x+y+1)²

Find the particular solution of (1+ x³)dy − x² ydx = satisfying the condition y(1) = 2.

Form the differential equation by eliminating the arbitrary constants A and B from y = Acos x + B sin x

Show that the function y=Acos2x-Bsin2x is a solution of the D.E y² +4y=0

Find the differential equation of the family of all non-vertical lines in a plane.

Find the differential equation of the family of all nonhorizontal lines in a plane.
Form the D.E to y² =a(b-x)(b+x) by eliminating a and b as its parameters.

Obtain the D.E of all circles of radius ‘r’

Part - C

Generic 5 - Answer All Question

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

Solve the Linear differential equation: (1 − x²) \(\frac{dy}{dx}\)− xy = 1

Find the equation of the curve whose slope is \(\frac{y-1}{x^2+x}\)and which passes through the point (1,0).

Solve the differential equation: (ydx-xdy)cot \(\left(\frac{x}{y}\right)\)=ny² dx

The velocity v , of a parachute falling vertically satisfies the equation \(v\frac{dv}{dx}=g\left(1-\frac{v^2}{k^2}\right)\), where g and k
are constants. If v and x are both initially zero, find v in terms of x.

Solve :x² dy+y(x+y)dx=0 given that y=1 when x=1.

Suppose a person deposits 10,000 Indian rupees in a bank account at the rate of 5% per annum
compounded continuously. How much money will be in his bank account 18 months later?
Find the differential equation for the family of all straight lines passing through the origin.

In a murder investigation, a corpse was found by a detective at exactly 8 p.m. Being alert, the
detective also measured the body temperature and found it to be 70^o F. Two hours later, the detective
measured the body temperature again and found it to be 60^o F. If the room temperature is 50^o F, and
assuming that the body temperature of the person before death was 98.6^o F, at what time did the
murder occur?
[log(2.43)=0.88789; log(0.5)=-0.69315]

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