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12th Business Maths Weekly Test - 1 ( Differential Equations )

St. Britto Hr. Sec. School - Madurai

12th Business Maths Weekly Test - 1 ( Differential Equations )-Aug 2020

Time : 2.30 Hours

Part - A

Multiple Choice Question - Answer All Question

The degree of the differential equation \(\frac { { d }^{ 4 }y }{ { dx }^{ 4 } } { -\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) }^{ 4 }+\frac { dy }{ dx } =3\)

1
2
3
4

The differential equation formed by eliminating A and B from y = e^−2x(A cos x + B sin x) is

y₂− 4y₁+ 5 = 0
y₂+ 4y – 5 = 0
y₂−4y₁−5= 0
y₂+ 4y₁+ 5 = 0

Which of the following is the homogeneous differential equation?

(3x−5)dx = (4y−1)dy
xy dx−(x³+y³)dy = 0
y²dx+(x²− xy − y²)dy = 0
(x²+y)dx = (y²+x)dy

The differential equation of x²+ y²= a²

xdy + ydx = 0
ydx – xdy = 0
xdx – ydx = 0
xdx + ydy = 0

The integrating factor of x \(\frac { dy }{ dx } \) - y = x² is

\(\frac {-1}{x}\)
\(\frac {1}{x}\)
log x
x

If sec² x is an integrating factor of the differential equation \(\frac { dy }{ dx } \) + Py Q then P =

2 tan x
sec x
cos² x
tan² x

A homogeneous differential equation of the form \(\frac { dy }{ dx } \) = f\(\left( \frac { y }{ x } \right) \) can be solved by making substitution,

y = v x
v = y x
x = v y
x = v

The complementary function of \(\frac { { d }^{ 3 }y }{ d{ x }^{ 3 } } -\frac { dy }{ dx } \) = 0 is

A + Be^x
(A + B)e^x
(Ax + B)e^x
Ae^x + B

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

y = sinx + 1
y = sinx - 2
y = cos x + c, c is an arbitrary constant
y = sin x + c, c is an arbitrary constant

The differential equation formed by eliminating a and b from y=ae^x+ be^−x is

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

Part - B

Generic 2 - Answer All Question

Find the order and degree of the following differential equations.
\(\frac { { d }^{ 3 }y }{ { dx }^{ 3 } } +3{ \left( \frac { dy }{ dx } \right) }^{ 3 }+2(\frac { dy }{ dx }) =0\)

Solve the following differential equations
\(\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } -4\frac { dy }{ dx } +4y=0\)

Solve the following differential equations (3D²+ D − 14)y=13e^2x

Form the differential equation by eliminating α and β from (x − α)² + (y − β)² = r²

Solve the following:
\(\frac { dy }{ dx } +ycosx=sinxcosx\).

A bank pays interest by continuous compounding , that is by treating the interest rate as the instantaneous rate of change of principal. A man invests Rs. 1,00,000 in the bank deposit which accures interest, 8% per year compounded continuously. How much will he get after 10 years.

(D²− 3D + 2)y = e^3x which shall vanish for x = 0 and for x = log 2

Solve: \(\frac { dy }{ dx } \) + e^x+ye^x= 0

Solve the following:
\(\frac { dy }{ dx } +\frac { y }{ x } ={ xe }^{ x }\)

Solve the following homogeneous differential equations.
\((x-y)\frac { dy }{ dx } =x+3y\).

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

Solve the following homogeneous differential equations.
\(x\frac { dy }{ dx } -y=\sqrt { { x }^{ 2 }+{ y }^{ 2 } } \)

Solve (x²+ y²)dx + 2xy dy = 0

Solve: log\(\left( \frac { dy }{ dx } \right) \) = ax + by

Find the differential equation of the family of all straight lines passing through the origin.

Find the order and degree of the following differential equations.
(2 - y'')²= y''² + 2y'

Find the differential equation of the following
x² + y² = a²

Form the differential equation having for its general solution y = ax²+ bx

Solve the following:
\(\frac { dy }{ dx } -\frac { y }{ x } =x\)

Find the order and degree of the following differential equations.
\(\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +y+{ \left( \frac { dy }{ dx } -\frac { { d }^{ 3 }y }{ dx^{ 3 } } \right) }^{ { 3 }/{ 2 } }\)

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