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JEE • MATHEMATICS • DIFFERENTIAL EQUATIONS

Differential equations practice · Medium

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391 Medium
391 of 452
Application +4 | -1
The solution of the differential equation xdy+ydx−1−x2y2dx=0x d y+y d x-\sqrt{1-x^2 y^2} d x=0 is
View answer and explanation
Correct option: B
xy=sin⁡(x+c)x y=\sin (x+c)
Option B (xy=sin⁡(x+c)x y=\sin (x+c)) is the correct solution, obtained by integrating the differential equation after recognizing the exact differential.
392 Medium
392 of 452
Analyzing +4 | -1
Family of curves y=Ax+A3y=A x+A^3 is represented by the differential equation of degree
View answer and explanation
Correct option: A
3
Option A (3) is correct because after eliminating the arbitrary constant A, the resulting differential equation has a degree of 3.
393 Medium
393 of 452
Application +4 | -1
If y=y(x)y=y(x) is the solution of the differential equation dydx+2y=sin⁡(2x),y(0)=34,\frac{d y}{d x}+2 y=\sin (2 x), y(0)=\frac{3}{4}, then y(π8)y\left(\frac{\pi}{8}\right) is equal to:
View answer and explanation
Correct option: B
e−π/4{e}^{-\pi / 4}
The correct answer is B (e−π/4{e}^{-\pi / 4}) because it is the value of y(π/8) obtained by solving the given differential equation with the initial condition y(0) = 3/4.
394 Medium
394 of 452
Application +4 | -1
Solution of the differential equation dydx=y(x−yln⁡y)x(xln⁡x−y)\frac{d y}{d x}=\frac{y(x-y \ln y)}{x(x \ln x-y)} is
View answer and explanation
Correct option: A
xln⁡x+yln⁡yxy=c\frac{x \ln x+y \ln y}{x y}=c
Option A (xln⁡x+yln⁡yxy=c\frac{{x \ln x+y \ln y}}{{x y}}=c) is correct because it is the solution obtained after integrating the given differential equation after rearrangement and simplification.
395 Medium
395 of 452
Application +4 | -1
If (2+sin⁡ x)dydx+(y+1)cos⁡x=0\left( {2 + \sin \,x} \right)\frac{{dy}}{{dx}} + \left( {y + 1} \right)\cos x = 0 and y(0)=1,then  y(π2)y\left( 0 \right) = 1,\mathrm{ then}\,\,y\left( {\frac{\pi }{2}} \right) is equal to
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(π2)y(\frac{\pi}{2}) calculated by solving the given differential equation with the initial condition.
396 Medium
396 of 452
Understanding +4 | -1
The differential equation of all parabolas whose axes are parallel to yy-axis, is
View answer and explanation
Correct option: A
d3ydx3=0\frac{{{d^3}y}}{{d{x^3}}} = 0
Option A (d3ydx3=0\frac{{{d^3}y}}{{d{x^3}}} = 0) is correct because the general equation of a parabola with its axis parallel to the y-axis is y=Ax2+Bx+Cy = Ax^2 + Bx + C, and its third derivative is zero.
397 Medium
397 of 452
Application +4 | -1
The solution of the differential equation (3xy+y2)dx+(x2+xy)dy=0\left( {3xy + {y^2}} \right)dx + \left( {{x^2} + xy} \right)dy = 0 is
View answer and explanation
Correct option: A
x2(2xy+y2)=c2{x^2}\left( {2xy + {y^2}} \right) = {c^2}
Option A (x2(2xy+y2)=c2{x^2}\left( {2xy + {y^2}} \right) = {c^2}) is correct because it is the solution obtained by solving the homogeneous differential equation.
398 Medium
398 of 452
Application +4 | -1
The equation of the curve whose tangent at any point (x,y)\left( {x,y} \right) makes an angle tan⁡−1(2x+3y){\tan ^{ - 1}}\left( {2x + 3y} \right) with xx-axis and which passes through (1,2) is
View answer and explanation
Correct option: A
6x+9y+2=26e3(x−1)6x + 9y + 2 = 26{e^{3\left( {x - 1} \right)}}
Option C (6x+9y−2=26e3(x−1)6x + 9y - 2 = 26{e^{3\left( {x - 1} \right)}}) is the correct solution to the differential equation given the initial condition (1,2).
399 Medium
399 of 452
Application +4 | -1
The solution of the differential equation d2ydx2=e−2x\frac{{{d^2}y}}{{d{x^2}}} = {e^{ - 2x}} is y=c1e−2x+c2x+x3y = {c_1}{e^{ - 2x}} + {c_2}x + {x_3}, where c1{c_1} is
View answer and explanation
Correct option: B
14\frac{1}{4}
Option B (14\frac{1}{4}) is correct because it is the coefficient of the exponential term in the particular solution obtained by integrating the given differential equation twice.
400 Medium
400 of 452
Application +4 | -1
The differential equation representing the family of curves y=xecxy = x{e^{cx}} (cc is a constant) is
View answer and explanation
Correct option: C
dydx=yx(1+log⁡yx)\frac{{dy}}{{dx}} = \frac{y}{x}\left( {1 + \log \frac{y}{x}} \right)
Option C is the correct answer because it represents the differential equation derived from the given family of curves by eliminating the constant 'c'.
401 Medium
401 of 452
Application +4 | -1
The solution of xdy−ydx+x2exdx=0x\mathrm{ }dy - y\mathrm{ }dx + {x^2}{e^x}dx = 0 is
View answer and explanation
Correct option: A
yx+ex=c\frac{y}{x} + {e^x} = c
Option A (yx+ex=c\frac{y}{x} + {e^x} = c) is correct because it is the solution to the given differential equation after rewriting it in an exact form and integrating.
402 Medium
402 of 452
Understanding +4 | -1
The order of the differential equation of all circles of radius rr, having centre on yy-axis and passing through the origin, is
View answer and explanation
Correct option: A
1
Option A (1) is correct because the differential equation of a one-parameter family of circles is of order one.
403 Medium
403 of 452
Application +4 | -1
For solution of differential equation xln⁡xdydx+y=x2ln⁡x, y(2)=2x \ln x \frac{d y}{d x}+y=x^2 \ln x, \, y(2)=2 then y(e)y(e) is equal to
View answer and explanation
Correct option: A
1+e241+\frac{e^2}{4}
Option A (−25\frac{-2}{5}) is correct because it represents the sum of the roots of the transformed equation f(1/t) = 0.
404 Medium
404 of 452
Application +4 | -1
If x=x(y)x = x\left( y \right) is the solution of the differential equation ydxdy=2x+y3(y+1)ey, x(1)=0;y\frac{{dx}}{{dy}} = 2x + {y^3}\left( {y + 1} \right){e^y},\,x\left( 1 \right) = 0; then x(e)x\left( e \right) is equal to:
View answer and explanation
Correct option: A
e3(ee−1){e^3}\left( {{e^e} - 1} \right)
Option A (e3(ee−1){e^3}({e^e} - 1)) is correct because it is the solution to the given differential equation with the specified initial condition, evaluated at y = e.
405 Medium
405 of 452
Application +4 | -1
Solution of the differential equation, sin⁡ydydx=cos⁡y(1−xcos⁡y)\sin y \frac{d y}{d x}=\cos y(1-x \cos y) is
View answer and explanation
Correct option: B
sec⁡y=x+1+cex\sec y=x+1+c e^{x}
Option B (sec⁡y=x+1+cex\sec y=x+1+c e^{x}) is correct because it is the solution obtained by solving the given differential equation using an integrating factor.

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406 Medium
406 of 452
Application +4 | -1
A function y=f(x)y=f(x) satisfying the differential equation dydx⋅sin⁡x−y⋅cos⁡x+sin⁡2xx2=0\frac{d y}{d x} \cdot \sin x-y \cdot \cos x+\frac{\sin ^2 x}{x^2}=0 is such that y→1y \rightarrow 1 as x→π4,x \rightarrow \frac{\pi}{4}, then Ltx→0f(x)=\underset{x \rightarrow 0}{\text{Lt}} f(x)=
View answer and explanation
Correct option: B
1
Option B (1) is correct because after solving the differential equation and applying the given condition, the limit of f(x) as x approaches 0 is found to be 1.
407 Medium
407 of 452
Application +4 | -1
If x2+y2=1{x^2} + {y^2} = 1, then
View answer and explanation
Correct option: B
yy′′+(y′)2+1=0yy'' + {\left( {y'} \right)^2} + 1 = 0
Option B (yy′′+(y′)2+1=0yy'' + {\left( {y'} \right)^2} + 1 = 0) is correct because it is the result of differentiating the given equation twice and eliminating x.
408 Medium
408 of 452
Application +4 | -1
An integrating factor of the differential equation x+dydx+ylog⁡x=xexx−12log⁡x,(x,0)x + \frac{{dy}}{{dx}} + y\log x = x{e^x}{x^{ - \frac{1}{2}\log x}},\left( {x,0} \right) is
View answer and explanation
Correct option: C
(e)(log⁡x)2{\left( {\sqrt e } \right)^{{{(\log x)}^2}}}
Option C ((e)(log⁡x)2( \sqrt{e} )^{{(\log x)}^2}) is correct because it is the integrating factor calculated using the formula e∫Pdxe^{\int P dx} for the given differential equation.
409 Medium
409 of 452
Application +4 | -1
The equation of the curve whose tangent at any point (x,y)(x, y) makes an angle tan⁡−1(2x+3y)\tan ^{-1}(2 x+3 y) with xx-axis and which passes through (1,2)(1,2) is:
View answer and explanation
Correct option: A
6x+9y+2=26e3(x−1)6 x+9 y+2=26 e^{3(x-1)}
Option A (6x+9y+2=26e3(x−1)6x+9y+2=26e^{3(x-1)}) is correct because it represents the solution to the given differential equation, obtained by integrating and applying the initial condition.
410 Medium
410 of 452
Application +4 | -1
The general solution of the differential equation (2x−y+1)dx+(2y−x+1)dy=0(2 x-y+1) d x+(2 y-x+1) d y=0 is
View answer and explanation
Correct option: B
x2+y2−xy+x+y=cx^2+y^2-x y+x+y=c
Option B (x2+y2−xy+x+y=cx^2+y^2-x y+x+y=c) is the correct general solution to the given differential equation, found by substituting x=X+hx=X+h and y=Y+ky=Y+k to simplify the equation.
411 Medium
411 of 452
Application +4 | -1
A function ff is differentiable on R\mathrm{R} and f(1)=1f(1)=1. The yy-intercept of the tangent at point P(x,y)\mathrm{P}(x, y) of curve y=f(x)y=f(x) is equal to the cube of the xx coordinate, then f(−3)=……f(-3)=\ldots \ldots.
View answer and explanation
Correct option: A
9
Option A (9) is correct because it is the value of the function f(x) at x = -3, obtained by solving the given differential equation and using the initial condition f(1) = 1.
412 Medium
412 of 452
Analyzing +4 | -1
The differential equation satisfied by the family of curves y=axcos⁡(1x+b)y = ax\cos \left( {\frac{1}{x} + b} \right), where aa and bb are parameters, is
View answer and explanation
Correct option: B
x4y2+y=0{x^4}{y_2} + y = 0
Option B (x4y2+y=0{x^4}{y_2} + y = 0) is correct because it is the differential equation obtained by differentiating the given equation twice and eliminating the parameters a and b.
413 Medium
413 of 452
Application +4 | -1
The solution of (y(1+x−1)+sin⁡y)dx+\left(y\left(1+x^{-1}\right)+\sin y\right) \mathrm{d} x+ (x+log⁡x+xcos⁡y)dy=0(x+\log x+x \cos y) \mathrm{d} y=0 is
View answer and explanation
Correct option: C
xy+ylog⁡x+xsin⁡y=Cx y+y \log x+x \sin y=C
Option C (xy+ylog⁡x+xsin⁡y=Cx y+y \log x+x \sin y=C) is correct because the given differential equation is exact and its solution is found by integrating the terms.
414 Medium
414 of 452
Application +4 | -1
Equation of the curve whose slope at the point (x,y)(x, y) is −(x+y)/x-(x+y) / x and which passes through the point (2,1)(2,1) is :
View answer and explanation
Correct option: C
x2+2xy=8x^2+2 x y=8
Option C (x2+2xy=8x^2+2 x y=8) is correct because it represents the solution to the given differential equation that passes through the point (2,1).
415 Medium
415 of 452
Application +4 | -1
If xy=x+y,dydx=x y=x+y, \frac{d y}{d x}=
View answer and explanation
Correct option: D
−1(x−1)2\frac{-1}{(x-1)^2}
Option D is correct because it is the result of differentiating the given equation implicitly and simplifying.
416 Medium
416 of 452
Application +4 | -1
The solution of the equation y−xdydx=a(y2+dydx)y - x\frac{{dy}}{{dx}} = a\left( {{y^2} + \frac{{dy}}{{dx}}} \right) is
View answer and explanation
Correct option: A
y=c(x+a)(1−ay)y = c\left( {x + a} \right)\left( {1 - ay} \right)
Option A (y=c(x+a)(1−ay)y = c\left( {x + a} \right)\left( {1 - ay} \right)) is correct because it is the solution obtained after separating variables and integrating the given differential equation.
417 Medium
417 of 452
Application +4 | -1
If ∫axty(t)dt=x2+y(x)\int_a^x t y(t) d t=x^2+y(x), then yy as a function of xx is
View answer and explanation
Correct option: A
y=2−(2+a2)ex2−a22y=2-\left(2+a^2\right) e^{\frac{x^2-a^2}{2}}
Option A, y=2−(2+a2)ex2−a22y=2-(2+a^2) e^{\frac{x^2-a^2}{2}}, is the correct function yy as a function of xx obtained by solving the given integral equation.
418 Medium
418 of 452
Application +4 | -1
The solution of (x dx+y dy)(x2+y2)+(x dy−y dx)(x \mathrm{~d} x+y \mathrm{~d} y)\left(x^2+y^2\right)+(x \mathrm{~d} y-y \mathrm{~d} x) =0=0 is
View answer and explanation
Correct option: D
x2+y22+tan⁡−1(xy)=C\frac{x^2+y^2}{2}+\tan ^{-1}\left(\frac{x}{y}\right)=C
Option D, x2+y22+tan⁡−1(xy)=C\frac{x^2+y^2}{2}+\tan ^{-1}\left(\frac{x}{y}\right)=C, is the correct solution. It is obtained by recognizing exact differentials and integrating.
419 Medium
419 of 452
Application +4 | -1
Integrating factor of (x+2y3)dydx=y2\left( {x + 2{y^3}} \right)\frac{{dy}}{{dx}} = {y^2} is
View answer and explanation
Correct option: A
e(1y){e^{\left( {\frac{1}{y}} \right)}}
Option A (e(1y){e^{\left( {\frac{{1}}{{y}}} \right)}}) is correct because it is the integrating factor found after rewriting the differential equation and applying the integrating factor formula.
420 Medium
420 of 452
Application +4 | -1
The solution of the differential equation (2x−10y3)dydx,y=0\left(2 x-10 y^3\right) \frac{d y}{d x}, y=0 is
View answer and explanation
Correct option: C
xy2−2y5+cx y^2-2 y^5+c
Option C (xy2−2y5+cx y^2-2 y^5+c) is the correct solution obtained by solving the differential equation using an integrating factor.
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