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

St. Britto Hr. Sec. School - Madurai

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

Time : 2.00 Hours

Part - A

Generic 2 - Answer All Question

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

\(\frac{d^2y}{dx^2}+5\frac{dy}{dx}+∫ydx = x^3\)
A differential equation, determine its order, degree (if exists)

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

Determine the order and degree (if exists) of the following differential equations:\(\frac{dy}{dx}\) = x + y + 5

Find value of m so that the function y = e^mx is a solution of the given differential equation.
y''− 5y' + 6y = 0

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

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

Determine the order and degree (if exists) of the following differential equations:

\(3\left(\frac{d^2y}{dx^2}\right)=\left[4+\left(\frac{dy}{dx}\right)^2\right]^\frac{3}{2}\)

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

Determine the order and degree (if exists) of the following differential equations:

\(\left(\frac{d^4y}{dx^4}\right)^3+\left(\frac{dy}{dx}\right)^7+6y=5cos3x\)

Solve: \(\frac{dy}{dx}=\frac{2x}{x^2+1}\)
Form the differential equation satisfied by are the straight lines in my-plane.

Form the D.E corresponding to y=e^mx by eliminating 'm'.

Solve the Linear differential equation:

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

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

Show that y = ae^-3x + b, where a and b are arbitary constants, is a solution of the differential equation

\(\frac{d^2y}{dx^2}+3\frac{dy}{dx}=0\)

Form the differential equation of all straight lines touching the circle x² + y² = r² .

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

Solve the differential equation:\(\frac{dy}{dx}\) = tan²(x + y)

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

Find the differential equations of the family of all the ellipses having foci on the y-axis and centre at
the origin.

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

\(\frac{d^2y}{dx^2}=xy+cos\left(\frac{dy}{dx}\right)\)

Solve: \(\frac{dy}{dx}+y=1\)

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

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

Find the differential equation corresponding to the family of curves represented by the equation y =
Ae^8x + Be^-8x , where A and B are arbitrary constants.

A curve passing through the origin has its slope e^x, Find the equation of the curve.

Part - B

Generic 5 - Answer All Question

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

Solve the Linear differential equation: (x + a) \(\frac{dy}{dx}\)− 2y = (x + a)⁴
Solve y ' = sin² (x − y + )1.

Solve the Linear differential equation: (x² + 1)\(\frac{d}{y}\)dx+2xy=\(\sqrt{x^2+4}\)

Solve (1+x³)\(\frac{dy}{dx}\)+6x²y=1+x² .

At 10.00 A.M. a woman took a cup of hot instant coffee from her microwave oven and placed it on a
nearby Kitchen counter to cool. At this instant the temperature of the coffee was 180^o F, and 10
minutes later it was 160^o F. Assume that constant temperature of the kitchen was 70^o F.
(i) What was the temperature of the coffee at 10.15A.M.?
(ii) The woman likes to drink coffee when its temperature is between 130^o F and 140^o F.between what
times should she have drunk the coffee?

Solve the differential equation: \(\left[x+y \quad cos\left(\frac{y}{x}\right)\right]\)dx=x cos\(\left(\frac{y}{x}\right)\)dy

Solve (2x + 3y)dx + ( y − x)dy = 0.

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

If F is the constant force generated by the motor of an automobile of mass M, its velocity is given by
M \(\frac{dV}{dt}\)=F-kV, where k is a constant. Express V in terms of t given that V = 0 when t = 0.

Solve \(\frac{dy}{dx}\)=\(\frac{x − y + 5}{2 ( x − y ) + 7}\).

Comment

Add Comment

InstaQB

Institutions

12th Standard English Medium

12th Standard Tamil Medium

11th Standard English Medium

11th Standard Tamil Medium

10th Standard English Medium

9th Standard English Medium

8th Standard English Medium

7th Standard English Medium

Class XII

Class XI

12Th Standard

11Th Standard

10Th Standard

9Th Standard

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

St. Britto Hr. Sec. School - Madurai

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

Part - A

Generic 2 - Answer All Question

A differential equation, determine its order, degree (if exists) \(\frac{d^2y}{dx^2}+5\frac{dy}{dx}+∫ydx = x^3\)

A differential equation, determine its order, degree (if exists) \(x^2\frac{d^2y}{dx^2}+\left[1+\left(\frac{dy}{dx}\right)^2\right]^\frac{1}{2}=0\)

Determine the order and degree (if exists) of the following differential equations:\(\frac{dy}{dx}\) = x + y + 5

Find value of m so that the function y = emx is a solution of the given differential equation. y''− 5y' + 6y = 0

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

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

Determine the order and degree (if exists) of the following differential equations: \(3\left(\frac{d^2y}{dx^2}\right)=\left[4+\left(\frac{dy}{dx}\right)^2\right]^\frac{3}{2}\)

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

Determine the order and degree (if exists) of the following differential equations: \(\left(\frac{d^4y}{dx^4}\right)^3+\left(\frac{dy}{dx}\right)^7+6y=5cos3x\)

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

Form the differential equation satisfied by are the straight lines in my-plane.

Form the D.E corresponding to y=emx by eliminating 'm'.

Solve the Linear differential equation: \(\frac{dy}{dx}=\frac{sin^2x}{1+x^3}-\frac{3x^2}{1+x^{3^y}}\)

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

Show that y = ae-3x + b, where a and b are arbitary constants, is a solution of the differential equation \(\frac{d^2y}{dx^2}+3\frac{dy}{dx}=0\)

Form the differential equation of all straight lines touching the circle x2 + y2 = r2 .

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

Solve the differential equation:\(\frac{dy}{dx}\) = tan2(x + y)

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

Find the differential equations of the family of all the ellipses having foci on the y-axis and centre at the origin.

A differential equation, determine its order, degree (if exists) \(\frac{d^2y}{dx^2}=xy+cos\left(\frac{dy}{dx}\right)\)

Solve: \(\frac{dy}{dx}+y=1\)

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

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

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

Find the differential equation corresponding to the family of curves represented by the equation y = Ae8x + Be-8x , where A and B are arbitrary constants.

A curve passing through the origin has its slope ex, Find the equation of the curve.

Part - B

Generic 5 - Answer All Question

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

Solve the Linear differential equation: (x + a) \(\frac{dy}{dx}\)− 2y = (x + a)4

Solve y ' = sin2 (x − y + )1.

Solve the Linear differential equation: (x2 + 1)\(\frac{d}{y}\)dx+2xy=\(\sqrt{x^2+4}\)

Solve (1+x3)\(\frac{dy}{dx}\)+6x2y=1+x2 .

Solve the differential equation: \(\left[x+y \quad cos\left(\frac{y}{x}\right)\right]\)dx=x cos\(\left(\frac{y}{x}\right)\)dy

Solve (2x + 3y)dx + ( y − x)dy = 0.

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

If F is the constant force generated by the motor of an automobile of mass M, its velocity is given by M \(\frac{dV}{dt}\)=F-kV, where k is a constant. Express V in terms of t given that V = 0 when t = 0.

Solve \(\frac{dy}{dx}\)=\(\frac{x − y + 5}{2 ( x − y ) + 7}\).

Comment

Other Study material

12th Maths Monthly Test - 2 ( Discrete Mathematics

12th Maths Monthly Test - 3 ( Discrete Mathematics

12th Maths Monthly Test - 1 ( Applications of Vect

12th Maths Monthly Test - 2 ( Applications of Vect

12th Maths Monthly Test - 3 ( Applications of Vect

12th Maths Monthly Test - 1 ( Ordinary Differentia

12th Maths Monthly Test - 2 ( Ordinary Differentia

12th Maths Monthly Test - 3 (Ordinary Differential

12th Maths Monthly Test - 1 (Probability Distribut

12th Maths Monthly Test - 2 ( Probability Distribu

12th Maths Monthly Test - 3 (Probability Distribut

12th Maths Monthly Test - 1 ( Discrete Mathematics

12th Maths Weekly Test -1 (Ordinary Differential E

12th Maths Weekly Test -2 (Ordinary Differential E

12th Maths Weekly Test -3 (Ordinary Differential E

Tamilnadu Stateboardszs - Class

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

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

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

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

Find value of m so that the function y = e^mx is a solution of the given differential equation.
y''− 5y' + 6y = 0

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

Determine the order and degree (if exists) of the following differential equations:

\(3\left(\frac{d^2y}{dx^2}\right)=\left[4+\left(\frac{dy}{dx}\right)^2\right]^\frac{3}{2}\)

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

Determine the order and degree (if exists) of the following differential equations:

\(\left(\frac{d^4y}{dx^4}\right)^3+\left(\frac{dy}{dx}\right)^7+6y=5cos3x\)

Form the D.E corresponding to y=e^mx by eliminating 'm'.

Solve the Linear differential equation:

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

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

Show that y = ae^-3x + b, where a and b are arbitary constants, is a solution of the differential equation

\(\frac{d^2y}{dx^2}+3\frac{dy}{dx}=0\)

Form the differential equation of all straight lines touching the circle x² + y² = r² .

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

Solve the differential equation:\(\frac{dy}{dx}\) = tan²(x + y)

Find the differential equations of the family of all the ellipses having foci on the y-axis and centre at
the origin.

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

\(\frac{d^2y}{dx^2}=xy+cos\left(\frac{dy}{dx}\right)\)

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

Find the differential equation corresponding to the family of curves represented by the equation y =
Ae^8x + Be^-8x , where A and B are arbitrary constants.

A curve passing through the origin has its slope e^x, Find the equation of the curve.

Solve the Linear differential equation: (x + a) \(\frac{dy}{dx}\)− 2y = (x + a)⁴

Solve y ' = sin² (x − y + )1.

Solve the Linear differential equation: (x² + 1)\(\frac{d}{y}\)dx+2xy=\(\sqrt{x^2+4}\)

Solve (1+x³)\(\frac{dy}{dx}\)+6x²y=1+x² .

If F is the constant force generated by the motor of an automobile of mass M, its velocity is given by
M \(\frac{dV}{dt}\)=F-kV, where k is a constant. Express V in terms of t given that V = 0 when t = 0.