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JEE • MATHEMATICS • DIFFERENTIAL EQUATIONS

Differential equations practice · Mixed difficulty

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421 Medium
421 of 975
Application +4 | -1
The solution of the equation dydx+ytan⁡x=xmcos⁡x\frac{d y}{d x}+y \tan x=x^m \cos x is
View answer and explanation
Correct option: A
(m+1)y=xm+1cos⁡x+c(m+1)cos⁡x(m+1) y=x^{m+1} \cos x+c(m+1) \cos x
Option A ((m+1)y=xm+1cos⁡x+c(m+1)cos⁡x(m+1) y=x^{m+1} \cos x+c(m+1) \cos x) is the correct solution obtained by solving the linear differential equation using an integrating factor.
422 Medium
422 of 975
Application +4 | -1
The differential equation of the family of the curves x2+y2−2ax=0{x^2} + {y^2} - 2ax = 0 is
View answer and explanation
Correct option: B
y2−x2=2xyy′{y^2} - {x^2} = 2xyy'
Option B, y2−x2=2xyy′{y^2} - {x^2} = 2xyy', is the correct differential equation because it is obtained by differentiating the given equation and eliminating the arbitrary constant 'a'.
423 Easy
423 of 975
Application +4 | -1
The equation of curves which intersect the hyperbola xy=4x y=4 at an angle π/2\pi / 2 is
View answer and explanation
Correct option: C
y2−x2=Cy^2-x^2=C
Option C (y2−x2=Cy^2-x^2=C) is correct because it represents the orthogonal trajectories to the given hyperbola.
424 Easy
424 of 975
Understanding +4 | -1
The second order differential equation is
View answer and explanation
Correct option: B
y′y′′+y=sin⁡xy'y'' + y = \sin x
Option B (y′y′′+y=sin⁡xy'y'' + y = \sin x) is correct because it contains a second-order derivative (y′′y'') and no derivative of higher order.
425 Medium
425 of 975
Application +4 | -1
If xdy−ydx+xcos⁡(log⁡ex)dx=0,y(1)=1x d y-y d x+x \cos \left(\log _e^x\right) d x=0, y(1)=1, then y(e)=y(e)=
View answer and explanation
Correct option: B
e(1−sin⁡1)e(1-\sin 1)
Option B (e(1-sin 1)) is correct because solving the differential equation and applying the initial condition y(1)=1 leads to y(e) = e(1-sin 1).
426 Medium
426 of 975
Analyzing +4 | -1
The degree of differential equation satisfying the relation 1+x2+1+y2=\sqrt{1+x^2}+\sqrt{1+y^2}= λ(x1+y2−y1+x2)\lambda\left(x \sqrt{1+y^2}-y \sqrt{1+x^2}\right) is
View answer and explanation
Correct option: A
1
Option A (1) is correct because after simplification and differentiation, the highest order derivative present is of order 1, and its power is also 1.
427 Medium
427 of 975
Application +4 | -1
Solution of the equation dydx=ex−y(ex−ey)\frac{d y}{d x}=e^{x-y}\left(e^x-e^y\right) is
View answer and explanation
Correct option: B
ey=(ex−1)+c(eex)−1e^y=\left(e^x-1\right)+c\left(e^{e^x}\right)^{-1}
Option B (ey=(ex−1)+c(eex)−1e^y=(e^x-1)+c(e^{e^x})^{-1}) is the correct solution to the given differential equation, obtained by transforming it into a linear form and applying the integrating factor.
428 Easy
428 of 975
Application +4 | -1
If ϕ(x)\phi(x) is a differentiable function, then the solution of the differential equation
dy+{y ϕ′(x)−ϕ(x)ϕ′(x)}dx=0d y+\left\{y \, \phi^{\prime}(x)-\phi(x) \phi^{\prime}(x)\right\} d x=0, is
View answer and explanation
Correct option: A
y={ϕ(x)−1}+Ce−ϕ(x)y=\{\phi(x)-1\}+C e^{-\phi(x)}
Option A (y={ϕ(x)−1}+Ce−ϕ(x)y=\{\phi(x)-1\}+C e^{-\phi(x)}) is the correct solution after solving the linear differential equation using the integrating factor method.
429 Easy
429 of 975
Application +4 | -1
Which of the following differential equation has y=xy=x as one of its particular solution?
View answer and explanation
Correct option: A
d2ydx2−x2dydx+xy=x\frac{d^2 y}{d x^2}-x^2 \frac{d y}{d x}+x y=x
Option A is correct because when y=xy=x is substituted into the differential equation, it satisfies the equation.
430 Easy
430 of 975
Application +4 | -1
If dydx+2ytan⁡x=sin⁡x, 0<x<π2\frac{{dy}}{{dx}} + 2y\tan x = \sin x,\,0 < x < \frac{\pi }{2} and y(π3)=0,y\left( {\frac{\pi }{3}} \right) = 0, then the maximum value of y(x)y(x) is:
View answer and explanation
Correct option: A
18\frac{1}{8}
Option A (1/8) is the correct maximum value of y(x) found by solving the differential equation and finding the maximum of the resulting function.
431 Medium
431 of 975
Application +4 | -1
Let the population of rats surviving at time tt be govemed by the differential equation dp(t)dt=0.5p(t)−450\frac{d p(t)}{d t}=0.5 p(t)-450. If p(0)=850p(0)=850, then at what time will they extinct ?
View answer and explanation
Correct option: A
2log⁡182 \log 18
Option A (2log⁡182 \log 18) is correct because it represents the time at which the rat population becomes zero, calculated by solving the given differential equation.
432 Medium
432 of 975
Application +4 | -1
The temperature T(t)T(t) of a body at time t=0t=0 is 160∘F160^{\circ} \mathrm{F} and it decreases continuously as per the differential equation dTdt=−K(T−80)\frac{d T}{d t}=-K(T-80), where KK is a positive constant. If T(15)=120∘FT(15)=120^{\circ} \mathrm{F}, then T(45)T(45) is equal to
View answer and explanation
Correct option: B
90∘F90^{\circ} \mathrm{F}
Option B (90∘F90^{\circ} \mathrm{F}) is correct because it is the temperature obtained by solving the given differential equation with the provided initial conditions and time values.
433 Easy
433 of 975
Application +4 | -1
The solution of the equation can be written as xy′−yx=tan⁡yx\frac{x y^{\prime}-y}{x}=\tan \frac{y}{x} is given by
View answer and explanation
Correct option: A
sin⁡yx=Cx\sin \frac{y}{x}=C x
The correct answer is A (sin⁡yx=Cx\sin \frac{{y}}{{x}}=C x) because it is the general solution to the given homogeneous differential equation.
434 Medium
434 of 975
Application +4 | -1
The solution of the differential equation dydx=xy+yxy+x\frac{{dy}}{{dx}} = \frac{{xy + y}}{{xy + x}} is
View answer and explanation
Correct option: D
y−x=log⁡(cxy)y - x = \log \left( {\frac{{cx}}{y}} \right)
Option D (y−x=log⁡(cxy)y - x = \log \left( {\frac{{cx}}{y}} \right)) is correct because it is the solution obtained by separating variables and integrating the given differential equation.
435 Easy
435 of 975
Application +4 | -1
If y=y(x)y=y(x) and 2sin⁡xy+1dydx=−cos⁡x\frac{2 \sin x}{y+1} \frac{d y}{d x}=-\cos x, y(0)=1y(0)=1, then y(π/2)y(\pi / 2) equal
View answer and explanation
Correct option: A
13\frac{1}{3}
Option A (13\frac{{1}}{{3}}) is correct because it is the value of y(π/2\pi/2) obtained by solving the differential equation with the given initial condition.

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436 Medium
436 of 975
Application +4 | -1
The real value of nn for which the substitution y=uny = {u^n} will transform the differential equation 2x4ydydx+y4=4x62{x^4}y\frac{{dy}}{{dx}} + {y^4} = 4{x^6} into a homogenous equation is
View answer and explanation
Correct option: C
32\frac{3}{2}
Option C (32\frac{3}{2}) is correct because it is the value of n that transforms the given differential equation into a homogeneous equation.
437 Medium
437 of 975
Application +4 | -1
The slope of the tangent at (x,y)\left( {x,y} \right) to a curve passing through a point (2,1)\left( {2,1} \right) is x2+y22xy\frac{{{x^2} + {y^2}}}{{2xy}}, then the equation of the curve is
View answer and explanation
Correct option: A
2(x2−y2)=3x2\left( {{x^2} - {y^2}} \right) = 3x
Option A (2(x2−y2)=3x2\left( {{x^2} - {y^2}} \right) = 3x) is correct because it is the equation of the curve obtained by solving the differential equation and applying the given point (2, 1).
438 Easy
438 of 975
Understanding +4 | -1
General solution of the differential equation ydx−xdyy2=0\frac{y d x-x d y}{y^2}=0 is
View answer and explanation
Correct option: D
y=cxy=c x
Option D (y=cxy=cx) is correct because the given differential equation simplifies to dyy=dxx\frac{{dy}}{{y}} = \frac{{dx}}{{x}}, and integrating both sides gives log⁡∣y∣=log⁡∣x∣+c\log |y| = \log |x| + c, which leads to y=cxy = cx.
439 Medium
439 of 975
Analyzing +4 | -1
If (cos⁡xcot⁡y−sin⁡xtan⁡y)dy+(cos⁡xln⁡cos⁡y−sin⁡xln⁡sin⁡y)dx;0<y<π2(\cos x \cot y-\sin x \tan y) d y+(\cos x \ln \cos y-\sin x \ln \sin y) d x ; 0 < y < \frac{\pi}{2}. If y(π2)=π4y\left(\frac{\pi}{2}\right)=\frac{\pi}{4}, then the value of y(0)y(0) is
View answer and explanation
Correct option: C
π4\frac{\pi}{4}
Option C is correct because solving the given differential equation and applying the initial condition yields a constant, and subsequently, y(0) = π/4.
440 Medium
440 of 975
Application +4 | -1
A curve passes through (1,2)(1,2). If the product of the slope of the tangent at any point (x,y)(x, y) of the curve and yy-coordinate of the point is equal to its xx-coordinate, then the curve is
View answer and explanation
Correct option: C
A rectangular hyperbola
Option C (A rectangular hyperbola) is correct because the equation derived from the given condition represents a rectangular hyperbola.
441 Medium
441 of 975
Application +4 | -1
The solution of diferential equation (xy2−ex13)dx−x2ydy=0\left(x y^2-e^{x^{\frac{1}{3}}}\right) d x-x^2 y d y=0 is
View answer and explanation
Correct option: B
3y2=2x2ex13+Cx23 y^2=2 x^2 e^{x^{\frac{1}{3}}}+C x^2
Option B (3y2=2x2ex13+Cx23 y^2=2 x^2 e^{x^{\frac{1}{3}}}+C x^2) is correct because it is the solution obtained after solving the given differential equation using an integrating factor.
442 Easy
442 of 975
Application +4 | -1
If dydx=y+3>0\frac{d y}{d x}=y+3 > 0 and y(0)y(0) and y(0)=2y(0)=2, y(log⁡2)y(\log 2) is equal to
View answer and explanation
Correct option: B
7
Option B (7) is correct because it is the value of y(log⁡2)y(\log 2) obtained by solving the differential equation and applying the given initial condition.
443 Easy
443 of 975
Application +4 | -1
The solution of differential equation xydydx=1+y21+x2(1+x+x2)\mathrm{xy} \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1+\mathrm{y}^2}{1+\mathrm{x}^2}\left(1+\mathrm{x}+\mathrm{x}^2\right) given that x=1,y=0\mathrm{x}=1, \mathrm{y}=0 is
View answer and explanation
Correct option: B
ℓn(1+y2x2)=2tan⁡−1x−π2\ell \mathrm{n} \left(\frac{1+\mathrm{y}^2}{\mathrm{x}^2}\right)=2 \tan ^{-1} \mathrm{x}-\frac{\pi}{2}
Option B (ln((1+y²)/x²)=2tan⁻¹x-π/2) is correct because solving the separable differential equation and applying the initial condition x=1, y=0 leads to this solution.
444 Easy
444 of 975
Application +4 | -1
The expression satisfying the differential equation (x2−1)dydx+2xy=1\left(x^2-1\right) \frac{d y}{d x}+2 x y=1 is
View answer and explanation
Correct option: C
(x2−1)y=x+c\left(x^2-1\right) y=x+c
Option C, (x2−1)y=x+c\left(x^2-1\right) y=x+c, is the correct solution because it satisfies the given differential equation after multiplying by the integrating factor.
445 Easy
445 of 975
Application +4 | -1
Solution of 2ysin⁡xdydx=2sin⁡xcos⁡x−y2cos⁡x,x=π2,y=12 y \sin x \frac{d y}{d x}=2 \sin x \cos x-y^2 \cos x, x=\frac{\pi}{2}, y=1 is given by
View answer and explanation
Correct option: A
y2=sin⁡xy^2=\sin x
Option A (y2=sin⁡xy^2=\sin x) is correct because it satisfies the given differential equation and the initial condition.
446 Easy
446 of 975
Application +4 | -1
The order and degree of the differential equation dydx−4dydx−7x=0\sqrt {\frac{{dy}}{{dx}}} - 4\frac{{dy}}{{dx}} - 7x = 0 are
View answer and explanation
Correct option: D
1 and 2
Option D (1 and 2) is correct because after squaring the equation to remove the square root, the highest order derivative is dydx\frac{dy}{dx} (order 1), and its power is 2 (degree 2).
447 Medium
447 of 975
Application +4 | -1
General solution of the equation dydx+2ytan⁡x=y2\frac{d y}{d x}+2 y \tan x=y^2 is
View answer and explanation
Correct option: C
cos⁡2x+y(x2+14sin⁡2x)=cy\cos ^2 x+y\left(\frac{x}{2}+\frac{1}{4} \sin 2 x\right)=c y
Option C (cos⁡2x+y(x2+14sin⁡2x)=cy\cos ^2 x+y(\frac{x}{2}+\frac{1}{4} \sin 2 x)=c y) is the correct general solution obtained by solving the Bernoulli differential equation.
448 Easy
448 of 975
Application +4 | -1
If y=(2πx−1)y = \left( {\frac{2}{\pi }x - 1} \right) cosec  xcosec\,\, x is the solution of the differential equation, dydx+p(x)y=2π   cosec  x,  0<x<π2\frac{{dy}}{{dx}} + p\left( x \right)y = \frac{2}{\pi }\,\,\,\mathrm{cosec }\,\,x,\,\,0 < x < \frac{\pi }{2} then the function p(x)p(x) is equal to
View answer and explanation
Correct option: A
cot  xcot \,\,x
Option A (cot xcot \, x) is correct because substituting the given solution into the differential equation and solving for p(x)p(x) yields cot xcot \, x.
449 Easy
449 of 975
Understanding +4 | -1
The order and the degree of d2ydx23=dydx\sqrt[3]{\frac{d^2 y}{d x^2}}=\sqrt{\frac{d y}{d x}} are and respectively .
View answer and explanation
Correct option: D
2,2
Option D (2,2) is correct because after eliminating the fractional powers, the order and degree are both 2.
450 Medium
450 of 975
Analyzing +4 | -1
The differential equation of the family of parabola with focus as the origin and the axis as xx-axis, is
View answer and explanation
Correct option: B
−y(dydx)2=2xdydx−y- y{\left( {\frac{{dy}}{{dx}}} \right)^2} = 2x\frac{{dy}}{{dx}} - y
Option B (−y(dydx)2=2xdydx−y- y{\left( {\frac{{dy}}{{dx}}} \right)^2} = 2x\frac{{dy}}{{dx}} - y) is the correct differential equation for the family of parabolas with focus at the origin and axis as the x-axis.
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