Mathematics - Differential Equations
Exam Duration: 45 Mins Total Questions : 30
The order and degree of the differential equation \(\frac { { d }^{ 2 }s }{ d{ r }^{ 2 } } +3{ \left( \frac { ds }{ dt } \right) }^{ 2 }\) = t \(\log { \left( \frac { { d }^{ 2 }s }{ d{ r }^{ 2 } } \right) } \), respectively are
- (a)
2, 2
- (b)
2, 1
- (c)
2, 3
- (d)
NONE OF THESE
The degree of the differential equation \({ \left( \frac { { d }^{ 3 }y }{ { dx }^{ 3 } } \right) }^{ 4/3 }+3\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +\frac { dy }{ dx } \) + 1 = 0, is
- (a)
3
- (b)
4
- (c)
2
- (d)
NOT DEFINED
A curve passes through the point \(\left( 1,\frac { \pi }{ 4 } \right) \) and its slope \(\frac{dy}{dx}\) at any point (x, y) is given by \(\frac { dy }{ dx } =\frac { y }{ x } -\cos ^{ 2 }{ \left( \frac { y }{ x } \right) } \) Then the equation of the curve, is
- (a)
y = \(\tan ^{ -1 }{ \left( \log { \frac { e }{ x } } \right) } \)
- (b)
y = x \(\tan ^{ -1 }{ \left( \log { \frac { x }{ e } } \right) } \)
- (c)
y = x \(\tan ^{ -1 }{ \left( \log { \frac { e }{ x } } \right) } \)
- (d)
y = \(\cot ^{ -1 }{ \left( \log { \frac { e }{ x } } \right) } \)
Solution of the equation \(\frac { xdx+ydy }{ { x }^{ 2 }+{ y }^{ 2 } } \)= k, is
- (a)
\({ x }^{ 2 }+{ y }^{ 2 }=c{ e }^{ 2kx }\)
- (b)
\(\tan ^{ -1 }{ \left( \frac { x }{ y } \right) } =kx+c\)
- (c)
\(\tan ^{ -1 }{ \left( \frac { x }{ y } \right) } =ky+c\)
- (d)
NONE OF THESE
The equation of the curve passing through the point (2, 1) and having subnormal as constant and of length 5 units at this points, is
- (a)
y2 = 10x - 19
- (b)
x2 = y2 + 3
- (c)
2y2 = x
- (d)
NONE OF THESE
The part of the normal between the point P(x, y) of a curve and the x-axis is constant k. Then the curve is
- (a)
a circle
- (b)
a parabola
- (c)
an ellipse
- (d)
a hyperbola
If \(L\frac { di }{ dt } +R\iota =E\), where L, R, E are constant and i = 0 when t = 0, then solution of the equation is
- (a)
\(i=\frac { E }{ R } \left( 1-{ e }^{ Rt/L } \right) \)
- (b)
\(i=\frac { E }{ R } \left( 1+{ e }^{ -Rt/L } \right) \)
- (c)
\(i=\frac { E }{ R } \left( 1-{ e }^{ -Rt/L } \right) \)
- (d)
\(i=\frac { E }{ R } \left( 1+{ e }^{ Rt/L } \right) \)
Observe the following columns
Column I | Column II |
A. Solutions of the differential equation \({ \left( \frac { dY }{ dX } \right) }^{ 2 }-\frac { dY }{ dX } \left( { e }^{ x }+{ e }^{ -x } \right) +1=0\) are given by | P. \(X+Y+2={ ce }^{ y }\) (where, c is arbitrary constant) |
B. Solutions of the differential equation (X+Y+1)dY = dX are given by | Q. In (X+Y+2)=c+Y (where, c is arbitrary constant) |
C. Solutions of the differential equation (X+Y+2)dX+(2X+2Y-1)dY = 0 are given by | R. Y+e-x = c (where, c is arbitrary constant) |
S. 2(X+Y+2)+5In(X+Y-3)=X+c (where, c is arbitrary constant. | |
T. Y - ex = c (where, c is arbitrary constant) |
- (a)
A B C RT P S - (b)
A B C QS P R - (c)
A B C T QR P - (d)
A B C S PS T
Solve the differential equation \(X\frac { dY }{ dX } =Y\left( \log { Y-\log { X+1 } } \right) \)
- (a)
\(\log { \left( \frac { y }{ x } \right) } ={ x }^{ 2 }C\)
- (b)
\(\log { \left( \frac { y }{ x } \right) } ={ 2x } C\)
- (c)
\(\log { \left( \frac { y }{ x } \right) } ={ x } C\)
- (d)
\(\log { \left( \frac { y }{ x } \right) } ={ y C}\)
If \(\frac { dY }{ dX } =Y+3>0\) and Y(0) = 2, then Y(log 2) is equal to
- (a)
5
- (b)
13
- (c)
- 2
- (d)
7
Consider the differential equation. \({ Y }^{ 2 }dX+\left( X-\frac { 1 }{ Y } \right) dY=0\) If Y(1) =1, then X is given by
- (a)
\(1-\frac { 1 }{ y } +\frac { { e }^{ 1/y } }{ e } \)
- (b)
\(4-\frac { 2 }{ y } -\frac { { e }^{ 1/y } }{ e } \)
- (c)
\(3-\frac { 1 }{ y } +\frac { { e }^{ 1/y } }{ e } \)
- (d)
\(1+\frac { 1 }{ y } -\frac { { e }^{ 1/y } }{ e } \)
The Order of the differential equation is the order of the highest derivative appearing in the equation and the Degree of a differential equation which can be written as polynomial in the derivatives in the degree of the derivative of the highest order occuring in it, after it has been expressed in a form free from radicals and fractions and if differential equation can not be written as a polynomial in the derivatives, then degree deos not defined but order defined.
The order and degree of the differential equation \(\frac { d^{ 2 }y }{ dx^{ 2 } } \) = cos\(\left( \frac { dy }{ dx } \right) \)+ xy are respecively
- (a)
2,1
- (b)
2,0
- (c)
2, infinite
- (d)
2, not defined
The largest value of c such that there exists a differential function h (x) for -c < x < c that is a solution of y1 = 1+ y2 with h(0) = 0 is
- (a)
\(2\pi\)
- (b)
\(\pi\)
- (c)
\(\frac{\pi}{2}\)
- (d)
\(\frac{\pi}{4}\)
The degree of the equation satisfying the relation \(\sqrt { 1+{ x }^{ 2 } } +\sqrt { 1+{ y }^{ 2 } } =\lambda \left( x\sqrt { 1+{ y }^{ 2 } } -y\sqrt { 1+{ x }^{ 2 } } \right) \) , is
- (a)
1
- (b)
2
- (c)
3
- (d)
none of these
The differential equation of family of curves whose tangent form an angle of π/4 with the hyperbola xy = c2 is
- (a)
\(\frac { dy }{ dx } =\frac { { x }^{ 2 }+{ c }^{ 2 } }{ { x }^{ 2 }-{ c }^{ 2 } } \)
- (b)
\(\frac { dy }{ dx } =\frac { { x }^{ 2 }-{ c }^{ 2 } }{ { x }^{ 2 }+{ c }^{ 2 } } \)
- (c)
\(\frac { dy }{ dx } =-\frac { { c }^{ 2 } }{ { x }^{ 2 } } \)
- (d)
none of these
If \(x\cos { \frac { y }{ x } \left( xdy+ydx \right) } =y\sin { \frac { y }{ x } \left( xdy-ydx \right) } \) , then
- (a)
\(\cos { \frac { y }{ x } } =cxy\)
- (b)
\(\sec { \frac { y }{ x } =cxy } \)
- (c)
x cos (xy) = cy
- (d)
x sec (xy) = cy
If \(\frac { dy }{ dx } =\frac { 2 }{ x+y } \), then x + y + 2 =
- (a)
cey
- (b)
ce y/2
- (c)
ce-y
- (d)
\(c{ e }^{ -\frac { y }{ 2 } }\)
If ydx - xdy + ln x dx = 0, y(1) = -1, then
- (a)
y + 1 + ln x = 0
- (b)
y + 1 + 2 ln x =0
- (c)
2(y + 1) + ln x = 0
- (d)
y + 1 - y ln x = 0
The solution of the differential equation \(\frac { x+y\frac { dy }{ dx } }{ y-x\frac { dy }{ dx } } ={ x }^{ 2 }+2{ y }^{ 2 }+\frac { { y }^{ 4 } }{ { x }^{ 2 } } \) is
- (a)
\(\frac { y }{ 4 } +\frac { 1 }{ { x }^{ 2 }+{ y }^{ 2 } } =c\)
- (b)
\(\frac { 2y }{ x } -\frac { 1 }{ { x }^{ 2 }+{ y }^{ 2 } } =c\)
- (c)
\(\frac { x }{ y } -\frac { 1 }{ { x }^{ 2 }+{ y }^{ 2 } } =c\)
- (d)
none of these
If \(\frac { dy }{ dx } \) = y sin 2x, y(0) = 1 then solution is
- (a)
\(y={ e }^{ \sin ^{ 2 }{ x } }\)
- (b)
\(y=\sin ^{ 2 }{ x } \)
- (c)
\(y=\cos ^{ 2 }{ x } \)
- (d)
\(y={ e }^{ \cos ^{ 2 }{ x } }\)
If \(\frac { xdy }{ dx } -y=\sqrt { { x }^{ 2 }+{ y }^{ 2 } } \), then
- (a)
\(x+\sqrt { { x }^{ 2 }+{ y }^{ 2 } } =c{ y }^{ 2 }\)
- (b)
\(\sqrt { { x }^{ 2 }+{ y }^{ 2 } } -y=c{ x }^{ 2 }\)
- (c)
\(\sqrt { { x }^{ 2 }+{ y }^{ 2 } } +y=c{ x }^{ 2 }\)
- (d)
\(\sqrt { { x }^{ 2 }+{ y }^{ 2 } }-x =c{ y }^{ 2 }\)
Solution of the differential equation \({ x }^{ 2 }\frac { dy }{ dx } .\cos { \frac { 1 }{ x } } -y\sin { \frac { 1 }{ x } } =-1\), where y ⇾ -1 as x ➝ ∞, is
- (a)
\(y=\sin { \frac { 1 }{ x } } -\cos { \frac { 1 }{ x } } \)
- (b)
\(y=\frac { x+1 }{ x\sin { \frac { 1 }{ x } } } \)
- (c)
\(y=\cos { \frac { 1 }{ x } } +\sin { \frac { 1 }{ x } } \)
- (d)
\(y=\frac { x+1 }{ x\cos { \frac { 1 }{ x } } } \)
The solution of \(\frac { dy }{ dx } +y={ e }^{ -x }\), y(0) = 0 is
- (a)
y = ex(x - 1)
- (b)
y = xe-x
- (c)
y = xex + 1
- (d)
y = (x + 1) e-x
Integrating factor of the differential equation \(\frac { dy }{ dx } +y\tan { x } -\sec { x } =0\) is
- (a)
cos x
- (b)
sec x
- (c)
ecosx
- (d)
esecx
Family y = Ax + A3 of curveswill correspond to a differential equation of order
- (a)
3
- (b)
2
- (c)
1
- (d)
not defined
The solution of the equation (2y - 1)dx - (2x + 3)dy = 0 is
- (a)
\(\frac { 2x-1 }{ 2y+3 } =k\)
- (b)
\(\frac { 2y+1 }{ 2x-3 } =k\)
- (c)
\(\frac { 2x+3 }{ 2y-1 } =k\)
- (d)
\(\frac { 2x-1 }{ 2y-1 } =k\)
The differential equation for which y = a cosx + b sin x is a solution, is
- (a)
\(\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +y=0\)
- (b)
\(\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } -y=0\)
- (c)
\(\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +\left( a+b \right) y=0\)
- (d)
\(\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +\left( a-b \right) y=0\)
The order and degree of the differential equation \({ \left( \frac { { d }^{ 3 }y }{ d{ x }^{ 3 } } \right) }^{ 2 }-3\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +2{ \left( \frac { dy }{ dx } \right) }^{ 4 }={ y }^{ 4 }\) are
- (a)
1, 4
- (b)
3, 4
- (c)
2, 4
- (d)
3, 2
Statement-I: The differential equation of all circles in a plane must be of order 3.
Statement-II: If three points are non-collinear, then only one circle always passing through these points.
- (a)
If both Statement-I and Statement-II are true and Statement-II is the correct explanation of Statement -1.
- (b)
If both Statement-I and Statement-II are true but Statement-II is not the correct explanation of Statement -1.
- (c)
If Statement-I is true but Statement-II is false.
- (d)
If Statement-I is false and Statement-II is true.