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JEE • MATHEMATICS • DIFFERENTIAL EQUATIONS

Differential equations practice · Mixed difficulty

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541 Medium
541 of 975
Application +4 | -1
Solution of the differential equation cos⁡xdy=y(sin⁡x−y)dx,0<x<π/2\cos x d y=y(\sin x-y) d x, 0 < x < \pi / 2 is
View answer and explanation
Correct option: C
sec⁡x=(tan⁡x+c)y\sec x=(\tan x+c) y
Option C (sec⁡x=(tan⁡x+c)y\sec x=(\tan x+c) y) is correct because it represents the solution to the given differential equation after applying appropriate substitutions and integration.
542 Medium
542 of 975
Application +4 | -1
General solution of the equation (xcos⁡x)dydx+y(xsin⁡x+cos⁡x)=1(x \cos x) \frac{d y}{d x}+y(x \sin x+\cos x)=1 is
View answer and explanation
Correct option: B
xysec⁡x=tan⁡x+cx y \sec x=\tan x+c
Option B (xysec⁡x=tan⁡x+cx y \sec x=\tan x+c) is correct because it represents the general solution of the given differential equation after finding the integrating factor and integrating.
543 Medium
543 of 975
Application +4 | -1
Solution of dydx=3x+y\frac{{dy}}{{dx}} = {3^{x + y}} is
View answer and explanation
Correct option: D
3x+3−y=c{3^x} + {3^{ - y}} = c
Option D (3x+3−y=c{3^x} + {3^{ - y}} = c) is correct because it represents the solution obtained after separating variables and integrating the given differential equation.
544 Medium
544 of 975
Application +4 | -1
The solution of differential equation dydx+y=1\frac{d y}{d x}+y=1 is
View answer and explanation
Correct option: A
y=1+ce−xy=1+c e^{-x}
Option A (y=1+ce−xy=1+c e^{-x}) is correct because it is the general solution to the given first-order linear differential equation.
545 Easy
545 of 975
Application +4 | -1
The trajectories orthogonal to x2+y2=2axx^2+y^2=2 a x is
View answer and explanation
Correct option: B
y=C(x2+y2)y=C\left(x^2+y^2\right)
The correct answer is B (y=C(x2+y2)y=C(x^2+y^2)) because it represents the family of curves orthogonal to the given trajectories, found by solving the differential equation.
546 Medium
546 of 975
Application +4 | -1
If xy=ex−yx^y=e^{x-y}, then dydx=\frac{d y}{d x}=
View answer and explanation
Correct option: A
log⁡x(1+log⁡x)2\frac{\log x}{(1+\log x)^2}
Option A (log⁡x(1+log⁡x)2\frac{\log x}{(1+\log x)^2}) is correct because differentiating yy with respect to xx after taking logarithms and simplifying yields this result.
547 Medium
547 of 975
Application +4 | -1
The equation of the curve passing through the point (1,π/4)(1, \pi / 4) and tangent at any point of which makes angle Tan⁡−1(yx−cos⁡2yx)\operatorname{Tan}^{-1}\left(\frac{y}{x}-\cos ^2 \frac{y}{x}\right) with the xx-axis, is
View answer and explanation
Correct option: C
y=xTan⁡−1(1−log⁡x)y=x \operatorname{Tan}^{-1}(1-\log x)
Option C (y=xTan⁡−1(1−log⁡x)y=x \operatorname{Tan}^{-1}(1-\log x)) is correct because it satisfies the given differential equation and passes through the point (1, π/4\pi/4).
548 Easy
548 of 975
Application +4 | -1
Solution to 9yy′+4x=09 y y^{\prime}+4 x=0 represent
View answer and explanation
Correct option: B
family of ellipses
Option B (family of ellipses) is correct because the solution to the given differential equation represents an ellipse.
549 Medium
549 of 975
Application +4 | -1
The degree of the differential equation satisfying 1−x2+1−y2=a(x−y)\sqrt {1 - {x^2}} + \sqrt {1 - {y^2}} = a\left( {x - y} \right) is
View answer and explanation
Correct option: A
1
Option A (1) is correct because after simplification and differentiation, the highest order derivative is of order 1.
550 Easy
550 of 975
Understanding +4 | -1
The degree and order of the differential equation of the family of all parabolas whose axis is xx - axis, are respectively.
View answer and explanation
Correct option: C
1,2
Option C (1,2) is correct because the degree of the differential equation is 1 and the order is 2.
551 Easy
551 of 975
Understanding +4 | -1
The order and degree of differential equation [1+(dydx)2]=d2ydx2\left[1+\left(\frac{d y}{d x}\right)^2\right]=\frac{d^2 y}{d x^2} are
View answer and explanation
Correct option: C
2,1
Option C (2,1) is correct because the order is the highest derivative (2) and the degree is the power of the highest derivative (1).
552 Medium
552 of 975
Application +4 | -1
The solution of differential equation (sin⁡x+cos⁡x)dy+(cos⁡x−sin⁡x)dx=0\left( {\sin x + \cos x} \right)dy + \left( {\cos x - \sin x} \right)dx = 0 is
View answer and explanation
Correct option: B
ey(sin⁡x+cos⁡x)=c{e^y}\left( {\sin x + \cos x} \right) = c
Option B (ey(sin⁡x+cos⁡x)=c{e^y}\left( {\sin x + \cos x} \right) = c) is correct because it is the solution obtained by integrating the given differential equation after separation of variables.
553 Easy
553 of 975
Application +4 | -1
Let a smooth curve y=f(x)y=f(x) be such that the slope of the tangent at any point (x,y)(x, y) on it is directly proportional to (−yx)\left(\frac{-y}{x}\right). If the curve passes through the points (1,2)(1,2) and (8,1)(8,1), then ∣y(18)∣\left|y\left(\frac{1}{8}\right)\right| is equal to
View answer and explanation
Correct option: B
4
Option B (4) is correct because solving the differential equation and applying the given conditions leads to the equation of the curve, which allows us to find the value of |y(1/8)|.
554 Medium
554 of 975
Application +4 | -1
Solution of dydx=xlog⁡x2+xsin⁡y+ycos⁡y\frac{{dy}}{{dx}} = \frac{{x\log {x^2} + x}}{{\sin y + y\cos y}} is
View answer and explanation
Correct option: A
ysin⁡y=x2log⁡x+cy\sin y = {x^2}\log x + c
Option A (ysin⁡y=x2log⁡x+cy\sin y = {x^2}\log x + c) is correct because it represents the solution obtained by separating variables and integrating the given differential equation.
555 Medium
555 of 975
Application +4 | -1
The differential equation ydydx+x=cy\frac{{dy}}{{dx}} + x = c represents
View answer and explanation
Correct option: D
A family of circles whose centres are on the xx-axis
Option D (A family of circles whose centres are on the xx-axis) is correct because the given differential equation can be integrated to obtain the equation of a circle with the center on the x-axis.

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556 Easy
556 of 975
Understanding +4 | -1
The solution of the equation d2ydx2=e−2x\frac{d^2 y}{d x^2}=e^{-2 x} is
View answer and explanation
Correct option: B
14e−2x+cx+d\frac{1}{4} e^{-2 x}+c x+d
Option B (14e−2x+cx+d\frac{{1}}{{4}} e^{-2 x}+c x+d) is the correct solution obtained by integrating the given second-order differential equation twice.
557 Easy
557 of 975
Application +4 | -1
The solution of differential equation yy′−x(y2x2+f(y2/x2)f′(y2/x2))y y^{\prime}-x\left(\frac{y^2}{x^2}+\frac{f\left(y^2 / x^2\right)}{f^{\prime}\left(y^2 / x^2\right)}\right) is
View answer and explanation
Correct option: A
f(y2/x2)−cx2f\left(y^2 / x^2\right)-c x^2
Option A is correct because it is the solution obtained by solving the given differential equation using the substitution method.
558 Medium
558 of 975
Analyzing +4 | -1
The differential equation of the system of all circles of radius rr in the xyxy-plane, is
View answer and explanation
Correct option: C
[1+(dydx)2]3=r2(d2ydx2)2{\left[ {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}} \right]^3} = {r^2}{\left( {\frac{{{d^2}y}}{{d{x^2}}}} \right)^2}
Option D ({\left[ {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}} \right]^3} = {r^2}{\left( {\frac{{{{d^2}y}}{{d{x^2}}}} \right)^2}{\left[ {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}} \right]^3} = {r^2}{\left( {\frac{{{{d^2}y}}{{d{x^2}}}} \right)^2}) is correct because it represents the differential equation derived from the general equation of a circle with radius r.
559 Easy
559 of 975
Application +4 | -1
The curves whose subtangents are proportional to the abscissas of the point of tangency (the proportionality factor is equal to kk ) is
View answer and explanation
Correct option: A
yk=Cxy^k=C x
Option A (yk=Cxy^k=C x) is correct because it is the solution obtained by integrating the differential equation formed from the given condition about subtangents.
560 Medium
560 of 975
Application +4 | -1
The solution of the differential equation xdydx=2y+x3exx\frac{{dy}}{{dx}} = 2y + {x^3}{e^x}, where y=0y = 0 when x=1x = 1, is
View answer and explanation
Correct option: C
y=x2(ex−e)y = {x^2}\left( {{e^x} - e} \right)
Option C (y=x2(ex−e)y = {x^2}\left( {{e^x} - e} \right)) is correct because it is the solution obtained by solving the linear differential equation and using the given initial condition.
561 Easy
561 of 975
Application +4 | -1
The solution of equation dydx=ax+qby+f\frac{d y}{d x}=\frac{a x+q}{b y+f} represents a circle only if
View answer and explanation
Correct option: B
a=−ba=-b
Option B (a=−ba=-b) is correct because for the given equation to represent a circle, the coefficients of x2x^2 and y2y^2 must be equal in magnitude and opposite in sign after integration.
562 Easy
562 of 975
Application +4 | -1
The differential equation for which sin⁡−1x+sin⁡−1y=c{\sin ^{ - 1}}x + {\sin ^{ - 1}}y = c is given by
View answer and explanation
Correct option: A
1−x2dy+1−y2dx=0\sqrt {1 - {x^2}} dy + \sqrt {1 - {y^2}} dx = 0
Option A (1−x2dy+1−y2dx=0\sqrt {1 - {x^2}} dy + \sqrt {1 - {y^2}} dx = 0) is correct because it is the differential equation obtained by differentiating the given equation and rearranging the terms.
563 Easy
563 of 975
Application +4 | -1
The differential equation of the family of curves y2=4a(x+a){y^2} = 4a\left( {x + a} \right), is
View answer and explanation
Correct option: B
y2(dydx)2+2xydydx−y2=0{y^2}{\left( {\frac{{dy}}{{dx}}} \right)^2} + 2xy\frac{{dy}}{{dx}} - {y^2} = 0
Option B (y2(dydx)2+2xydydx−y2=0{y^2}{\left( {\frac{{dy}}{{dx}}} \right)^2} + 2xy\frac{{dy}}{{dx}} - {y^2} = 0) is correct because it is the differential equation obtained by eliminating the parameter 'a' from the given family of curves.
564 Medium
564 of 975
Application +4 | -1
The general solution of the differential equation dydx+1+cos⁡2y1−cos⁡2x=0\frac{{dy}}{{dx}} + \frac{{1 + \cos 2y}}{{1 - \cos 2x}} = 0 is given by
View answer and explanation
Correct option: B
tan⁡y−cot⁡x=c\tan y - \cot x = c
Option B (tan⁡y−cot⁡x=c\tan y - \cot x = c) is correct because it is the general solution obtained by separating variables and integrating the given differential equation.
565 Medium
565 of 975
Application +4 | -1
Family of curves y=ex(Acos⁡x+Bsin⁡x)y=e^x(A \cos x+B \sin x), represents the differential equation
View answer and explanation
Correct option: B
d2ydx2=2dydx−2y\frac{d^2 y}{d x^2}=2 \frac{d y}{d x}-2 y
Option B (d2ydx2=2dydx−2y\frac{{d^2 y}}{{d x^2}}=2 \frac{{d y}}{{d x}}-2 y) is correct because differentiating the given family of curves twice and eliminating A and B leads to this differential equation.
566 Easy
566 of 975
Application +4 | -1
Solution of the differential equation x(dydx)2+2xydydx+y=0x\left(\frac{d y}{d x}\right)^2+2 \sqrt{x y} \frac{d y}{d x}+y=0, is
View answer and explanation
Correct option: D
x+y=a\sqrt{x}+\sqrt{y}=\sqrt{a}
Option D (x+y=a\sqrt{x}+\sqrt{y}=\sqrt{a}) is the correct solution obtained by recognizing the perfect square and integrating the resulting separable equation.
567 Medium
567 of 975
Application +4 | -1
If x3dy+xydx=x2dy+2ydx;y(2)=ex^3 d y+x y d x=x^2 d y+2 y d x ; y(2)=e and x>1x>1, then y(4)y(4) is equal to
View answer and explanation
Correct option: B
32e\frac{3}{2} \sqrt{e}
Option B (−(d2ydx2)(dydx)−3-( \frac{d^2 y}{d x^2})( \frac{d y}{d x})^{-3}) is correct because it is the correct expression for the second derivative of x with respect to y, derived using the chain rule.
568 Medium
568 of 975
Application +4 | -1
Integral curve satisfying y′=x2+y2x2−y2,y(1)=2y' = \frac{{{x^2} + {y^2}}}{{{x^2} - {y^2}}},y\left( 1 \right) = 2 has the slope at the point (1,0) of the curve is equal to
View answer and explanation
Correct option: C
1
Option C (1) is correct because the slope of the integral curve at the point (1,0) is found by substituting x=1 and y=0 into the derivative equation.
569 Easy
569 of 975
Application +4 | -1
Suppose y=y(x)y=y(x) satisfies the differential equation ydx+y2dy=xdyy d x+y^2 d y=x d y. If y(x)>0∀x∈Ry(x) > 0 \forall x \in \mathbf{R} and y(1)=1y(1)=1 then y(−3)y(-3) equals
View answer and explanation
Correct option: C
3
Option C (3) is correct because it is the value of y(-3) obtained by solving the given differential equation and applying the initial condition y(1)=1.
570 Medium
570 of 975
Application +4 | -1
The solution of (x+y+1)dydx=1\left( {x + y + 1} \right)\frac{{dy}}{{dx}} = 1 is
View answer and explanation
Correct option: C
x=−(y+2)+ceyx = - \left( {y + 2} \right) + c{e^y}
Option C, x=−(y+2)+ceyx = - \left( {y + 2} \right) + c{e^y}, is the correct solution because it satisfies the given differential equation after solving it as a linear differential equation in terms of x.
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