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JEE • MATHEMATICS • DIFFERENTIAL EQUATIONS

Differential equations practice · Mixed difficulty

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871 Medium
871 of 975
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.
872 Medium
872 of 975
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.
873 Easy
873 of 975
Application +4 | -1
Let the function x(t)&y(t)x(t) \& y(t) satisfy the differential equation dxdt+ax=0\frac{d x}{d t}+a x=0, dydt+by=0\frac{d y}{d t}+b y=0. If x(0)=2,y(0)=1x(0)=2, y(0)=1 and x(1)y(1)=32\frac{x(1)}{y(1)}=\frac{3}{2} then x(t)=y(t)x(t)=y(t) for t=t=
View answer and explanation
Correct option: B
log⁡4/32\log _{4 / 3} 2
Option B (log⁡4/32\log _{4 / 3} 2) is correct because solving the given differential equations and using the initial conditions and the ratio x(1)/y(1) allows us to find the value of t for which x(t) = y(t).
874 Easy
874 of 975
Application +4 | -1
The solution of ydx−xdy+3x2y2ex3dx=0y\mathrm{ }dx - x\mathrm{ }dy + 3{x^2}{y^2}{e^{{x^3}}}dx = 0 is
View answer and explanation
Correct option: A
xy+ex3=C\frac{x}{y} + {e^{{x^3}}} = C
Option A (xy+ex3=C\frac{x}{y} + {e^{{x^3}}} = C) is correct because it is the result of integrating the given differential equation after rewriting it in a suitable form.
875 Easy
875 of 975
Application +4 | -1
The integrating factor of the differential equation dydx−1−exy\frac{d y}{d x}-1-e^x y is
View answer and explanation
Correct option: B
eexe^{e^x}
Option B (eexe^{e^x}) is correct because it is the integrating factor obtained by integrating the coefficient of y in the given differential equation.
876 Easy
876 of 975
Application +4 | -1
If a curve y=f(x),y = f\left( x \right), passing through the point (1,2), is the solution of the differential equation, 2x2dy=(2xy+y2)dx,2{x^2}dy = \left( {2xy + {y^2}} \right)dx, then f(12)f\left( {\frac{1}{2}} \right) is equal to
View answer and explanation
Correct option: A
11+log⁡e2\frac{1}{{1 + {{\log }_e}2}}
Option A (11+log⁡e2\frac{{1}}{{{{1 + {{\log }_e}2}}}}) is correct because it is the value of the function at x=1/2 obtained by solving the given homogeneous differential equation and using the initial condition.
877 Medium
877 of 975
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.
878 Medium
878 of 975
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.
879 Easy
879 of 975
Remembering +4 | -1
The degree of the differential equation (d3ydx3)4+(d2ydx2)5+dydx+y=0\left(\frac{d^3 y}{d x^3}\right)^4+\left(\frac{d^2 y}{d x^2}\right)^5+\frac{d y}{d x}+y=0 is
View answer and explanation
Correct option: B
4
Option B (4) is correct because the degree of a differential equation is the highest power of the highest order derivative, which is 4 in this case.
880 Medium
880 of 975
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.
881 Medium
881 of 975
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.
882 Easy
882 of 975
Application +4 | -1
The differential equation d2ydx2=2\frac{{{d^2}y}}{{d{x^2}}} = 2 represents
View answer and explanation
Correct option: B
A parabola whose axis is parallel to yy-axis
Option B (A parabola whose axis is parallel to yy-axis) is correct because integrating the given differential equation twice results in the equation of a parabola with its axis parallel to the y-axis.
883 Easy
883 of 975
Understanding +4 | -1
The order and degree of the differential equation (1+4dydx)23=4d2ydx2{\left( {1 + 4\frac{{dy}}{{dx}}} \right)^{\frac{2}{3}}} = 4\frac{{{d^2}y}}{{d{x^2}}} are respectively
View answer and explanation
Correct option: C
2,32,3
Option C (2,32,3) is correct because after raising both sides to the power of 3, the highest order derivative is d2ydx2\frac{{{d^2}y}}{{d{x^2}}} (order 2) and its power is 3 (degree 3).
884 Medium
884 of 975
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.
885 Medium
885 of 975
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.

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886 Easy
886 of 975
Understanding +4 | -1
The degree of the differential equations x=1+(dydx)+12!(dydx)2+13!(dydx)3+…x = 1 + \left( {\frac{{dy}}{{dx}}} \right) + \frac{1}{{2!}}{\left( {\frac{{dy}}{{dx}}} \right)^2} + \frac{1}{{3!}}{\left( {\frac{{dy}}{{dx}}} \right)^3} + \ldots
View answer and explanation
Correct option: C
1
Option C (1) is correct because after simplification, the highest order derivative has a power of 1.
887 Easy
887 of 975
Application +4 | -1
The solution of dydx+y=e−x,y(0)=0\frac{{dy}}{{dx}} + y = {e^{ - x}},y\left( 0 \right) = 0, is
View answer and explanation
Correct option: B
y=xe−xy = x{e^{ - x}}
Option B (y=xe−xy = x{e^{ - x}}) is correct because it satisfies the given differential equation and the initial condition y(0)=0y(0) = 0.
888 Hard
888 of 975
Analyzing +4 | -1
The solution of the differential equation {1x−y2(x−y)2}dx+{x2(x−y)2−1y}dy=0\left\{ {\frac{1}{x} - \frac{{{y^2}}}{{{{\left( {x - y} \right)}^2}}}} \right\}dx + \left\{ {\frac{{{x^2}}}{{{{\left( {x - y} \right)}^2}}} - \frac{1}{y}} \right\}dy = 0 is
View answer and explanation
Correct option: A
In ∣xy∣+xy(x−y)=c\left| {\frac{x}{y}} \right| + \frac{{xy}}{{\left( {x - y} \right)}} = c
Option A (In ∣xy∣+xy(x−y)=c\left| {\frac{x}{y}} \right| + \frac{{xy}}{{\left( {x - y} \right)}} = c) is correct because it represents the solution obtained by integrating the given differential equation.
889 Medium
889 of 975
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.
890 Medium
890 of 975
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).
891 Medium
891 of 975
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.
892 Hard
892 of 975
Application +4 | -1
For the function y=Bx2y={Bx}^2 to be the solution of differential equation (dydx)3−15x2dydx−2xy=0\left(\frac{d y}{d x}\right)^3-15 x^2 \frac{d y}{d x}-2 x y=0, the value of BB is_____ , given that B≠0{B} \neq 0.
View answer and explanation
Correct option: A
2
Option A (6) is correct because substituting the given condition y=2 when x=1 into the general solution yields the constant term c = 6.
893 Easy
893 of 975
Application +4 | -1
Let y=f(x)y=f(x) be the solution of the differential equation y(x+1)dx−x2dy=0,y(1)=ey(x+1) d x-x^{2} d y=0, y(1)=e. Then lim⁡x→0f(x)\lim\limits_{x \to 0} f(x) is equal to
View answer and explanation
Correct option: B
0
Option B (0) is correct because the limit of the function f(x)f(x) as xx approaches 0 is 0, as derived from the solution of the differential equation.
894 Medium
894 of 975
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.
895 Easy
895 of 975
Application +4 | -1
The solution curve of the differential equation (1+e−x)(1+y2)dydx=y2\left(1+e^{-x}\right)\left(1+y^2\right) \frac{d y}{d x}=y^2, which passes through th point (0,1)(0,1) is
View answer and explanation
Correct option: D
y2=1+y(log⁡e(1+ex2))y^2=1+y\left(\log _e\left(\frac{1+e^x}{2}\right)\right)
Option D (y2=1+y(log⁡e(1+ex2))y^2=1+y\left(\log _e\left(\frac{1+e^x}{2}\right)\right)) is correct because it is the solution to the given differential equation that satisfies the initial condition (0,1).
896 Easy
896 of 975
Application +4 | -1
If y=xln⁡∣cx∣y=\frac{x}{\ln |c x|} (where cc is an arbitrary constant) is the general solution of the differential equation dydx=yx+ϕ(xy)\frac{d y}{d x}=\frac{y}{x}+\phi\left(\frac{x}{y}\right), then the function ϕ(xy)\phi\left(\frac{x}{y}\right) is
View answer and explanation
Correct option: D
−y2x2-\frac{y^2}{x^2}
Option D, −y2x2-\frac{y^2}{x^2}, is the correct function ϕ(xy)\phi(\frac{x}{y}) obtained by differentiating the given solution and substituting it into the differential equation.
897 Medium
897 of 975
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.
898 Easy
898 of 975
Understanding +4 | -1
Which of the following equations is a linear equation of order 3 ?
View answer and explanation
Correct option: C
xd3y dx3+d2y dx2=exx \frac{\mathrm{d}^3 y}{\mathrm{~d} x^3}+\frac{\mathrm{d}^2 y}{\mathrm{~d} x^2}=e^x
Option C (xd3y dx3+d2y dx2=exx \frac{\mathrm{d}^3 y}{\mathrm{~d} x^3}+\frac{\mathrm{d}^2 y}{\mathrm{~d} x^2}=e^x) is correct because it is a linear differential equation of order 3, where the dependent variable and its derivatives appear only to the first power and are not multiplied together.
899 Easy
899 of 975
Application +4 | -1
The degree of homogeneous function f(x,y)=x3−y3x2+y2f(x, y)=\frac{x^3-y^3}{x^2+y^2} is ...... .
View answer and explanation
Correct option: A
1
Option A (1) is correct because the function can be expressed as x times a function of y/x, indicating a homogeneous function of degree 1.
900 Medium
900 of 975
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.
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