JEE Main Mathematics - Differential Equations
Exam Duration: 60 Mins Total Questions : 30
The differential equation representing the family of curves given by the equation y = Aex + Be-x + C, is
- (a)
\(\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } =y\)
- (b)
\(\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } =\frac { dy }{ dx } \)
- (c)
\(\frac { { d }^{ 3 }y }{ d{ x }^{ 3 } } =\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } \)
- (d)
\({ \left( \frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } \right) }^{ 2 }+y=0\)
The differential equation of all circles of radius a, is
- (a)
\(\left[ 1+{ \left( \frac { dy }{ dx } \right) }^{ 2 } \right] ={ a }^{ 2 }{ \left( \frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } \right) }^{ 2 }\)
- (b)
\({ \left[ 1+{ \left( \frac { dy }{ dx } \right) }^{ 2 } \right] }^{ 3 }={ a }^{ 2 }{ \left( \frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } \right) }^{ 2 }\)
- (c)
\({ \left( 1+\frac { dy }{ dx } \right) }^{ 3 }={ a }^{ 2 }\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } \)
- (d)
NONE OF THESE
The solution of the differential equation \(\frac { dy }{ dx } =\frac { y }{ x } +\tan { \left( \frac { y }{ x } \right) } \), is
- (a)
\(\cos { \left( \frac { y }{ x } \right) } =cx\)
- (b)
\(\sin { \left( \frac { y }{ x } \right) } =cx\)
- (c)
\(\sin { \left( \frac { y }{ x } \right) } =cy\)
- (d)
\(\tan { \left( \frac { y }{ x } \right) } =cx\)
The equation of the curve passing through the origin and satisfying the differential equation \(\left( 1+{ x }^{ 2 } \right) \frac { dy }{ dx } +2xy=4{ x }^{ 2 }\), is
- (a)
\(\left( 1+{ x }^{ 2 } \right) y={ x }^{ 3 }\)
- (b)
\(2\left( 1+{ x }^{ 2 } \right) y=3{ x }^{ 3 }\)
- (c)
\(3\left( 1+{ x }^{ 2 } \right) y=4{ x }^{ 3 }\)
- (d)
NONE OF THESE
The solution of the equation \(\frac { dy }{ dx } +x\sin ^{ 2 }{ y={ x }^{ 3 }\cos ^{ 2 }{ y } } \), is
- (a)
\(y=\frac { 1 }{ 2 } \left( { x }^{ 2 }-1 \right) +c{ e }^{ { x }^{ 2 } }\)
- (b)
\(\tan { y=\left( { x }^{ 2 }-1 \right) +c{ e }{ x }^{ 2 } } \)
- (c)
\(\tan { y=\left( { x }^{ 2 }-1 \right) +c{ e }^{ { -x }^{ 2 } } } \)
- (d)
NONE OF THESE
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)
\(i=\frac { E }{ R } \left( 1+{ e }^{ Rt/L } \right) \)
The solution of differential equation \(\left( 1+{ y }^{ 2 } \right) \tan ^{ -1 }{ XdX+ } 2y\left( 1+{ X }^{ 2 } \right) dy=0\) is
- (a)
\(\frac { { \left( \tan ^{ -1 }{ x } \right) }^{ 2 } }{ 2 } +\log { \left| 1+{ y }^{ 2 } \right| } =C\)
- (b)
\(\frac { { \left( \tan ^{ -1 }{ x } \right) }^{ 2 } }{ 2 } -\log { \left| 1+{ y }^{ 2 } \right| } =C\)
- (c)
\(\frac { { \left( \tan ^{ -1 }{ x } \right) }^{ 2 } }{ 2 } -2\log { \left| 1+{ y }^{ 2 } \right| } =C\)
- (d)
\(\frac { { \left( \tan ^{ -1 }{ x } \right) }^{ 2 } }{ 2 } +5\log { \left| 1+{ y }^{ 2 } \right| } =C\)
The degree of the differential equation associated with \(\sqrt { 1+{ X }^{ 2 } } +\sqrt { 1+{ Y }^{ 2 } } \\ =\quad A\left( X\sqrt { 1+{ Y }^{ 2 } } -Y\sqrt { 1+{ X }^{ 2 } } \right) \) is
- (a)
1
- (b)
3
- (c)
4
- (d)
None of these
The solution of differential equation \(\frac { dY }{ dX } \left( { X }^{ 2 }{ Y }^{ 3 }+XY \right) =1\) is
- (a)
\(\frac { 1 }{ x } =\left( 2-{ y }^{ 2 } \right) +C{ e }^{ { -y }^{ 2 }/2 }\)
- (b)
\({ e }^{ { y }^{ 2 }/2 }\left( \frac { 1-2x }{ x } -{ y }^{ 2 } \right) =C\)
- (c)
\(\frac { 2 }{ x } =1-{ y }^{ 3 }+{ e }^{ { -y }^{ 2 }/3 }\)
- (d)
None of the above
If \(X\frac { dY }{ dX } =Y\left( \log { Y } -\log { X } +1 \right) \), then the solution of the equation is
- (a)
\(\log { \left( \frac { x }{ y } \right) } =Cy\)
- (b)
\(\log { \left( \frac { y }{ x} \right) } =Cx\)
- (c)
\(x\log { \left( \frac { y }{ x } \right) } =Cy\)
- (d)
\(y\log { \left( \frac { x }{ y } \right) } =Cx\)
The differential equation representing the family of the curves y2 = 2c ( x + √c )where c is a positive parameter, is of
- (a)
order 1, degree 3
- (b)
order 2, degree 2
- (c)
order 3, degree 3
- (d)
order 4, degree 4
The integrating factor of the differential equation \(\frac{dy}{dx}(xlog_{e}x)+y\) = 2logex is given by
- (a)
x
- (b)
ex
- (c)
loge x
- (d)
loge ( Ioge x )
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 particular solution of In \(\left( \frac { dy }{ dx } \right) \) = 3x + 4y, y (0) = 0 is
- (a)
3e3x + 4e-4y = 7
- (b)
3e4y - 4e-3x = 7
- (c)
4e3x x + 3e-4y = 7
- (d)
none of these
If the slope of tangent to the curve is maximum at x = 1 and curve has a minimum value 1 at x = 0, then the curve which also satisfies the equation y'''= 4x - 3 is
- (a)
\(y+2x+\frac { dy }{ dx } =0\)
- (b)
\(y=1+\frac { { x }^{ 2 } }{ 2 } -\frac { { x }^{ 3 } }{ 2 } +\frac { { x }^{ 4 } }{ 6 } \)
- (c)
y = 1 + x + x2 + x3
- (d)
none of these
The solution of the equation \(\frac{dy}{dx}\) - 3y = sin 2x is
- (a)
\({ ye }^{ -3x }=-\frac { 1 }{ 13 } { e }^{ -3x }(2cos2x+3sin2x)+c\)
- (b)
\(y=-\frac { 1 }{ 13 } (2cos2x+3sin2x)+ce^{ 3x }\)
- (c)
\(y=\{ -1/\sqrt { 13 } )cos(2x-tan^{ -1 }(3/2))+ce^{ 3x }\)
- (d)
\(y=\{ -1/\sqrt { 13 } )sin(2x+tan^{ -1 }(2/3))+ce^{ 3x }\)
The solution of y' (x2y3 + xy ) = 1 is (where c is arbitrary constant)
- (a)
\(\frac { 1 }{ x } =2-{ y }^{ 2 }+{ ce }^{ -y^{ 2/2 } }\)
- (b)
\(e^{ y^{ 2/2 } }\left( \frac { 1-2x }{ x } +{ y }^{ 2 } \right) =c\)
- (c)
the solution of an equation which is reducible to linear equation
- (d)
2/x = 1 - y2 +e-y/2
The differential equation having solution as y = 17ex + ae-x is
- (a)
y'' - x = 0
- (b)
y'' - y = 0
- (c)
y' - y = 0
- (d)
y' - x = 0
The general solution of the differential equation y(x2y + exdy)dx - exdy = 0, is
- (a)
x3y - 3ex = Cy
- (b)
x3y + 3ex = 3Cy
- (c)
y3x - 3ey = Cx
- (d)
y3x + 3ey = Cx
The solution of \(\frac { xdy }{ dx } =y\left( \ln { y } -\ln { x } +1 \right) \) is
- (a)
\(x\ln { \frac { y }{ x } =cy } \)
- (b)
\(\quad y\ln { \frac { x }{ y } =cx } \)
- (c)
\(\ln { \frac { x }{ y } =cy } \)
- (d)
\(\ln { \frac { y }{ x } =cx } \)
If \(\frac { 2dy }{ dx } =\frac { x }{ y } -1\) then
- (a)
(x + y)2 (2x - y) = c
- (b)
(x - y)2 (2y - x) = c
- (c)
(x - y)2 (2x - y) = c
- (d)
None of these
Solution of differential equation (x2 - 2x + 2y2) dx + 2xy dy = 0 is
- (a)
y2 = 2x - \(\frac { 1 }{ 4 } { x }^{ 2 }+\frac { c }{ { x }^{ 2 } } \)
- (b)
y2 = \(\frac { 2 }{ 3 } x-{ x }^{ 2 }+\frac { c }{ { x }^{ 2 } } \)
- (c)
y2 = \(\frac { 2 }{ 3 } x-\frac { { x }^{ 2 } }{ 4 } +\frac { c }{ { x }^{ 2 } } \)
- (d)
none of these
If e-x(A cos x + B sinx), then y is a solution of
- (a)
\(\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +2\frac { dy }{ dx } =0\)
- (b)
\(\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } -2\frac { dy }{ dx } +2y=0\)
- (c)
\(\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +2\frac { dy }{ dx }+2y =0\)
- (d)
\(\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +2y =0\)
Solution of diffeential equation xdy - ydx = 0 represents
- (a)
a rectangular hyperbola
- (b)
parabola whose vertex is at origin
- (c)
straight line passing thrrough origin
- (d)
a circle whose centre is at origin
The solution of the differential equation \(\frac { dy }{ dx } =\frac { 1+{ y }^{ 2 } }{ 1+{ x }^{ 2 } } \) is
- (a)
y = tan-1 x
- (b)
y - x = k(1 + xy)
- (c)
x = tan-1y
- (d)
tan(xy) = k
the solution of \(x\frac { dy }{ dx } +y={ e }^{ x }\) is
- (a)
\(y=\frac { { e }^{ x } }{ x } +\frac { k }{ x } \)
- (b)
y = xex + cx
- (c)
y = xex + k
- (d)
\(x=\frac { { e }^{ y } }{ y } +\frac { k }{ y } \)
The differential equation of the family of curves x2 + y2 - 2ay = 0, where a is arbitrary constant, is
- (a)
\(\left( { x }^{ 2 }-{ y }^{ 2 } \right) \frac { dy }{ dx } =2xy\)
- (b)
\(2\left( { x }^{ 2 }+{ y }^{ 2 } \right) \frac { dy }{ dx } =xy\)
- (c)
\(2\left( { x }^{ 2 }-{ y }^{ 2 } \right) \frac { dy }{ dx } =xy\)
- (d)
\(\left( { x }^{ 2 }+{ y }^{ 2 } \right) \frac { dy }{ dx } =2xy\)
General solution of \(\frac { dy }{ dx } \) + y tanx = sec x is
- (a)
y secx = tanx + c
- (b)
y tanx = secx + c
- (c)
tanx = y tanx + c
- (d)
x secx = tany + c