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JEE • MATHEMATICS • DIFFERENTIAL EQUATIONS

Differential equations practice · Mixed difficulty

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31 Medium
31 of 975
Analyzing +4 | -1
The differential equation of all parabolas having their axis of symmetry coinciding with the axis of XX, is
View answer and explanation
Correct option: A
yd2ydx2+(dydx)2=0y\frac{{{d^2}y}}{{d{x^2}}} + {\left( {\frac{{dy}}{{dx}}} \right)^2} = 0
Option A (yd2ydx2+(dydx)2=0y\frac{{{d^2}y}}{{d{x^2}}} + {\left( {\frac{{dy}}{{dx}}} \right)^2} = 0) is correct because it represents the differential equation of all parabolas with their axis of symmetry coinciding with the x-axis.
32 Easy
32 of 975
Application +4 | -1
The integrating factor of differential equation dydx+ytan⁡x−sec⁡x=0\frac{d y}{d x}+y \tan x-\sec x=0 is
View answer and explanation
Correct option: B
sec⁡x\sec x
Option B (sec x) is correct because it is the integrating factor obtained by solving the differential equation after identifying P(x) as tan x.
33 Medium
33 of 975
Application +4 | -1
The solution y′=1−2y−4x1+y+2xy^{\prime}=\frac{1-2 y-4 x}{1+y+2 x} is given by
View answer and explanation
Correct option: B
(y+12(y+2x)2)=x+C\left(y+\frac{1}{2}(y+2 x)^2\right)=x+C
The correct answer is B (y + 1/2(y+2x)2) = x+C because it is the solution obtained by substituting u = y + 2x and integrating the resulting separable equation.
34 Medium
34 of 975
Application +4 | -1
The solution of the differential equation 2xdydx−y=0;y(1)=22 x \frac{d y}{d x} - y=0 ; y(1)=2 represents ...... .
View answer and explanation
Correct option: B
A parabola
Option B (A parabola) is correct because the solution to the differential equation is of the form y2=kxy^2 = kx, which represents a parabola.
35 Medium
35 of 975
Application +4 | -1
The solution of the differential equation xx2+y2dy=(yx2+y2−1)dx\frac{x}{{{x^2} + {y^2}}}dy = \left( {\frac{y}{{{x^2} + {y^2}}} - 1} \right)dx, is
View answer and explanation
Correct option: C
y=xtan⁡(C−x)y = x\tan \left( {C - x} \right)
Option C (y=xtan⁡(C−x)y = x\tan (C - x)) is correct because it is the solution obtained after integrating the given differential equation using trigonometric substitution.
36 Easy
36 of 975
Application +4 | -1
The solution of the differential equation (x+y)2dydx=a2{\left( {x + y} \right)^2}\frac{{dy}}{{dx}} = {a^2} is
View answer and explanation
Correct option: D
None of these
Option D (None of these) is correct because the solution obtained through separation of variables and integration does not match any of the given options.
37 Easy
37 of 975
Application +4 | -1
Let the slope of the tangent to a curve y=f(x)y = f\left( x \right) at (x,y)\left( {x,y} \right) be given by 2  tan⁡x(cos⁡x−y).2\,\,\tan x\left( {\cos x - y} \right). If the curve passes through the point (π4,0),\left( {\frac{\pi }{4},0} \right), then the value of ∫0π/2ydx\int\limits_0^{\pi /2} y dx is equal to:
View answer and explanation
Correct option: B
2−π22 - \frac{\pi }{{\sqrt 2 }}
Option B (2 - π/√2) is correct because it is the value of the definite integral of y dx from 0 to π/2, where y is the solution to the given differential equation.
38 Easy
38 of 975
Application +4 | -1
Let y=y(x)y = y\left( x \right) be the solution of the differential equation x(1−x2)dydx+(3x2y−y−4x3)=0, x>1,x\left( {1 - {x^2}} \right)\frac{{dy}}{{dx}} + \left( {3{x^2}y - y - 4{x^3}} \right) = 0,\,x > 1, with y(2)=−2.y\left( 2 \right) = - 2. Then y(3)y\left( 3 \right) is equal to
View answer and explanation
Correct option: A
–18
Option A (–18) is correct because it is the value of y(3) obtained by solving the given differential equation with the initial condition y(2) = -2.
39 Easy
39 of 975
Application +4 | -1
The general solution of the differential equation dydx+2xy=x2\frac{d y}{d x}+\frac{2}{x} y=x^2, is
View answer and explanation
Correct option: B
y=cx−2+x35y=c x^{-2}+\frac{x^3}{5}
Option B (y=cx−2+x35y=c x^{-2}+\frac{{x^3}}{{5}}) is correct because it is the general solution obtained by solving the given linear differential equation using the integrating factor method.
40 Medium
40 of 975
Application +4 | -1
If ey+xy=e{e^y} + xy = e, the ordered pair (dydx,d2ydx2)\left( {\frac{{dy}}{{dx}},\frac{{{d^2}y}}{{d{x^2}}}} \right) at x=0 is equal to
View answer and explanation
Correct option: B
(−1e,1e2)\left( { - \frac{1}{e},\frac{1}{{{e^2}}}} \right)
Option B ((−1e,1e2)\left( { - \frac{{1}}{e},\frac{{1}}{{{{e}^2}}}} \right)) is correct because it is the ordered pair of the first and second derivatives of the given equation evaluated at x=0.
41 Medium
41 of 975
Analyzing +4 | -1
Number of straight lines which satisfy the differential equation dydx+x(dydx)2−y=0\frac{d y}{d x}+x\left(\frac{d y}{d x}\right)^2-y=0 is
View answer and explanation
Correct option: B
2
Option B (2) is correct because the differential equation represents a family of straight lines, and solving for the constants yields two possible straight lines.
42 Easy
42 of 975
Application +4 | -1
Let y=y (x)y = y\,(x) be the solution of the differential equation (x−x3)dy=(y+yx2−3x4)dx, x>2(x - {x^3})dy = (y + y{x^2} - 3{x^4})dx,\,x > 2. If y(3)=3y(3) = 3, then y (4)y\,(4) is equal to
View answer and explanation
Correct option: B
12
Option B (12) is correct. The solution involves solving a differential equation with an initial condition and then evaluating the solution at x=4.
43 Medium
43 of 975
Application +4 | -1
The differential equation of all circles passing through the origin and having their centres on the xx-axis is
View answer and explanation
Correct option: C
y2=x2+2xydydx{y^2} = {x^2} + 2xy\frac{{dy}}{{dx}}
Option C (y2=x2+2xydydx{y^2} = {x^2} + 2xy\frac{{dy}}{{dx}}) is correct because it represents the differential equation derived from the equation of circles passing through the origin with centers on the x-axis.
44 Easy
44 of 975
Application +4 | -1
If ydydx=x[y2x2+ϕ(y2x2)ϕ′(y2x2)],  x>0,  ϕ>0y\frac{{dy}}{{dx}} = x\left[ {\frac{{{y^2}}}{{{x^2}}} + \frac{{\phi \left( {\frac{{{y^2}}}{{{x^2}}}} \right)}}{{\phi '\left( {\frac{{{y^2}}}{{{x^2}}}} \right)}}} \right],{\mkern 1mu} {\mkern 1mu} x > 0,{\mkern 1mu} {\mkern 1mu} \phi > 0 and y(1)=−1y(1)=-1 then ϕ(y24)\phi \left( {\frac{{{y^2}}}{4}} \right) is equal to
View answer and explanation
Correct option: B
4  ϕ  (1)4\,\,\phi \,\,(1)
Option B (4  ϕ  (1)4\,\,\phi \,\,(1)) is correct because it is the solution obtained after solving the given differential equation and applying the initial condition.
45 Easy
45 of 975
Application +4 | -1
The integrating factor of the differential equation xdydx−y=2x2x \frac{d y}{d x}-y=2 x^2 is
View answer and explanation
Correct option: C
1/x1 / x
Option C (1/x1/x) is correct because after rewriting the equation in standard form, the integrating factor is calculated as e∫Pdxe^{\int P dx}.

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46 Medium
46 of 975
Application +4 | -1
The differential equation y5x+y−xdydx=0y^5 x+y-x \frac{d y}{d x}=0 has the general solution ......
View answer and explanation
Correct option: B
x55+14(xy)4=c\frac{x^5}{5}+\frac{1}{4}\left(\frac{x}{y}\right)^4=c
Option B (x55+14(xy)4=c\frac{{x^5}}{{5}} + \frac{{1}}{{4}}(\frac{{x}}{{y}})^4 = c) is correct because it is the result of solving the given differential equation by using a suitable substitution and integration.
47 Easy
47 of 975
Application +4 | -1
A solution of the differential equation (dydx)2−xdydx+y=0,\left(\frac{d y}{d x}\right)^2-x \frac{d y}{d x}+y=0, is
View answer and explanation
Correct option: C
y=2x−4y=2 x-4
Option C (y=2x−4y=2x-4) is correct because it satisfies the given differential equation when substituted.
48 Easy
48 of 975
Application +4 | -1
If a curve passes through the origin and the slope of the tangent to it at any point (x, y)\left( {x,\,y} \right) is (x2−4x+y+8)x−2\frac{{\left( {{x^2} - 4x + y + 8} \right)}}{{x - 2}}, then this curve also passes through the point
View answer and explanation
Correct option: D
(5, 5)\left( {5,\,5} \right)
Option D (5,5) is correct because the curve passes through this point after solving the differential equation.
49 Easy
49 of 975
Application +4 | -1
The slope of a curve at any point is the reciprocal of twice the ordinate at the point and it passes through the point (4, 3). The equation of the curve is
View answer and explanation
Correct option: C
y2=x+5{y^2} = x + 5
Option C (y2=x+5{y^2} = x + 5) is correct because it satisfies the given differential equation and passes through the point (4, 3).
50 Hard
50 of 975
Application +4 | -1
If dydx=y+xtan⁡yxx\frac{{dy}}{{dx}} = \frac{{y + x\tan \frac{y}{x}}}{x}, then sin⁡yx\sin \frac{y}{x} is equal to
View answer and explanation
Correct option: B
cxcx
Option B (cxcx) is correct because it is the result of solving the homogeneous differential equation using the substitution method.
51 Easy
51 of 975
Application +4 | -1
The solution of the differential equation (x2−yx2)dydx+y2+xy2=0\left( {{x^2} - y{x^2}} \right)\frac{{dy}}{{dx}} + {y^2} + x{y^2} = 0 is
View answer and explanation
Correct option: A
log⁡(xy)=1x+1y+c\log \left( {\frac{x}{y}} \right) = \frac{1}{x} + \frac{1}{y} + c
Option A (log⁡(xy)=1x+1y+c\log \left( {\frac{x}{y}} \right) = \frac{1}{x} + \frac{1}{y} + c) is correct because it is the solution obtained by separating variables and integrating the given differential equation.
52 Easy
52 of 975
Application +4 | -1
Let y=y(x)y=y(x) be the solution curve of the differential equation dydx+(2x2+11x+13x3+6x2+11x+6)y=(x+3)x+1,x>−1\frac{d y}{d x}+\left(\frac{2 x^2+11 x+13}{x^3+6 x^2+11 x+6}\right) y=\frac{(x+3)}{x+1}, x>-1, which passes through the point (0,1)(0,1). Then y(1)y(1) is equal to
View answer and explanation
Correct option: B
32\frac{3}{2}
Option B (32\frac{{3}}{{2}}) is correct because after solving the linear differential equation and applying the initial condition, y(1) evaluates to this value.
53 Medium
53 of 975
Application +4 | -1
The solution of the differential equation (e−2xx−yx)dxdy=1,x≠0\left(\frac{e^{-2 \sqrt{x}}}{\sqrt{x}}-\frac{y}{\sqrt{x}}\right) \frac{d x}{d y}=1, x \neq 0 is ...... -
View answer and explanation
Correct option: A
ye2x=2x+cy e^{2 \sqrt{x}}=2 \sqrt{x}+c
Option A (ye2x=2x+cy e^{2 \sqrt{x}}=2 \sqrt{x}+c) is the correct solution, derived by using an integrating factor after rearranging the equation.
54 Medium
54 of 975
Application +4 | -1
The solution to the differential equation dydx=e3x−2y+x2e−2y\frac{d y}{d x}=e^{3 x-2 y}+x^2 e^{-2 y} is
View answer and explanation
Correct option: B
3e2y=2(e3x+x3)+c3 {e}^{2 {y}}=2\left({e}^{3 {x}}+{x}^3\right)+{c}
Option B (3e2y=2(e3x+x3)+c3 {e}^{2 {y}}=2\left({e}^{3 {x}}+{x}^3\right)+{c}) is correct because separating variables and integrating leads to the given solution.
55 Medium
55 of 975
Application +4 | -1
The solution of the differential equation dydx+2ycot⁡x=3x2cosec⁡2x\frac{d y}{d x}+2 y \cot x=3 x^2 \operatorname{cosec}^2 x is
View answer and explanation
Correct option: A
ysin⁡2x=x3+cy \sin ^2 x=x^3+c
Option A (ysin⁡2x=x3+cy \sin ^2 x=x^3+c) is correct because it is the solution to the given differential equation after finding the integrating factor and integrating.
56 Medium
56 of 975
Application +4 | -1
The solution of the differential equation x(x−y)dydx=y(x+y)x\left( {x - y} \right)\frac{{dy}}{{dx}} = y\left( {x + y} \right), is
View answer and explanation
Correct option: A
xy+log⁡(xy)=c\frac{x}{y} + \log \left( {xy} \right) = c
Option A (xy+log⁡(xy)=c\frac{x}{y} + \log \left( {xy} \right) = c) is correct because it is the solution obtained by substituting y=vxy = vx and integrating the resulting equation.
57 Easy
57 of 975
Application +4 | -1
Let ff be a real-valued on R\mathbf{R} such that f(1)=1f(1)=1. If the yy-intercept of the tangent at any point {(x,y)\{(x, y) on the curve y=f(x)y=f(x) is equal to cube of the abscissa of PP, then the value of f(−3)f(-3) is equal to
View answer and explanation
Correct option: D
9
Option D (9) is correct because it is the value of f(-3) obtained by solving the given differential equation and applying the initial condition f(1)=1.
58 Medium
58 of 975
Application +4 | -1
The solution of the differential equation dydx=(4x+y+1)2\frac{{dy}}{{dx}} = {\left( {4x + y + 1} \right)^2}, is
View answer and explanation
Correct option: D
(4x+y+1)=2tan⁡(2x+c)\left( {4x + y + 1} \right) = 2\tan \left( {2x + c} \right)
Option D ((4x+y+1)=2tan⁡(2x+c)(4x + y + 1) = 2\tan (2x + c)) is correct because it is the solution of the given differential equation after substitution and integration.
59 Easy
59 of 975
Analyzing +4 | -1
The order of the differential equation of all tangent lines to the parabola y=x2y = {x^2} is
View answer and explanation
Correct option: A
1
Option A (1) is correct because the differential equation of all tangent lines to the parabola y=x2y = x^2 is of order 1.
60 Medium
60 of 975
Application +4 | -1
If y(0)=1y(0)=1, then the solution of the differential equation x3dydx−x2y=−y4cos⁡xx^3 \frac{d y}{d x}-x^2 y=-y^4 \cos x is given by
View answer and explanation
Correct option: B
x3=3y3sin⁡xx^3=3 y^3 \sin x
Option B (x3=3y3sin⁡xx^3=3 y^3 \sin x) is the correct solution to the given differential equation, obtained by transforming it into a linear form and applying the initial condition.
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