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JEE • MATHEMATICS • DIFFERENTIAL EQUATIONS

Differential equations practice · Easy

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211 Easy
211 of 494
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.
212 Easy
212 of 494
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.
213 Easy
213 of 494
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.
214 Easy
214 of 494
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.
215 Easy
215 of 494
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.
216 Easy
216 of 494
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.
217 Easy
217 of 494
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.
218 Easy
218 of 494
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.
219 Easy
219 of 494
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.
220 Easy
220 of 494
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.
221 Easy
221 of 494
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.
222 Easy
222 of 494
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).
223 Easy
223 of 494
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.
224 Easy
224 of 494
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.
225 Easy
225 of 494
Application +4 | -1
If dydx+ytan⁡x=sin⁡2x\frac{d y}{d x}+y \tan x=\sin 2 x and y(0)=1y(0)=1, then y(π)y(\pi) is equal to:
View answer and explanation
Correct option: C
-5
Option C (-5) is correct because solving the differential equation and applying the initial condition yields y(π) = -5.

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226 Easy
226 of 494
Application +4 | -1
An integrating factor of the differential equation xdydx+ylog⁡x=xexx−12log⁡x,(x>0)x\frac{{dy}}{{dx}} + y\log x = x{e^x}{x^{ - \frac{1}{2}\log x}},(x > 0), is
View answer and explanation
Correct option: C
(e)(log⁡x)2{\left( {\sqrt e } \right)^{{{\left( {\log x} \right)}^2}}}
Option C ((e)(log⁡x)2{\left( {\sqrt e } \right)^{{{\left( {\log x} \right)}^2}}}) is correct because it is the integrating factor obtained by solving the integral of the coefficient of y after rewriting the differential equation in standard form.
227 Easy
227 of 494
Understanding +4 | -1
The order and degree of the differential equation ρ=[1+(dydx)2]3/2d2ydx2\rho = \frac{{{{\left[ {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}} \right]}^{3/2}}}}{{\frac{{{d^2}y}}{{d{x^2}}}}} are respectively
View answer and explanation
Correct option: A
2,22,2
Option A (2,22,2) is correct because after squaring both sides to remove the fractional exponent, the highest order derivative is d2ydx2\frac{{d^2y}}{{dx^2}} (order 2) and its power is 2 (degree 2).
228 Easy
228 of 494
Application +4 | -1
Let y=y(x)y = y\left( x \right) be the solution of the differential equation, dydx+y=xlog⁡ex, (x>1).\frac{{dy}}{{dx}} + y = x{\log _e}x,\,\left( {x > 1} \right). If 2y(2)=log⁡e 4−1,2y\left( 2 \right) = {\log _e}\,4 - 1, then y(e)y(e) is equal to
View answer and explanation
Correct option: C
e4\frac{e}{4}
Option C (e4\frac{e}{4}) is correct because it's the value of y(e) obtained by solving the given differential equation and using the initial condition.
229 Easy
229 of 494
Application +4 | -1
The solution of dydx+y=ex\frac{{dy}}{{dx}} + y = {e^x} is
View answer and explanation
Correct option: C
2yex=e2x+c2y{e^x} = {e^{2x}} + c
Option C (2yex=e2x+c2y{e^x} = {e^{2x}} + c) is correct because it is the solution to the linear differential equation obtained by finding the integrating factor.
230 Easy
230 of 494
Application +4 | -1
The solution of the differential equation dydx+1−y21−x2=0\frac{d y}{d x}+\sqrt{\frac{1-y^2}{1-x^2}}=0, is
View answer and explanation
Correct option: B
sin⁡−1x+sin⁡−1y=C\sin ^{-1} x+\sin ^{-1} y=C
Option B (sin⁡−1x+sin⁡−1y=C\sin^{-1} x + \sin^{-1} y = C) is correct because it is the result of separating variables and integrating the given differential equation.
231 Easy
231 of 494
Understanding +4 | -1
The order of the differential equation
[1+5(dydx)2]3/2=11(d2ydx2)5\left[1+5\left(\frac{d y}{d x}\right)^2\right]^{3 / 2}=11\left(\frac{d^2 y}{d x^2}\right)^5 is
View answer and explanation
Correct option: B
2
Option B (2) is correct because the order of a differential equation is the highest order derivative present in the equation, which is the second derivative in this case.
232 Easy
232 of 494
Application +4 | -1
If y=f(x)y=f(x) passing through (1,2)(1,2) satisfies the differential equation y(1+xy)dx−xdy=0y(1+x y) d x-x d y=0, then
View answer and explanation
Correct option: A
f(x)=2x2−x2f(x)=\frac{2 x}{2-x^{2}}
Option A is correct because solving the differential equation and applying the initial condition yields the given function.
233 Easy
233 of 494
Application +4 | -1
In the particular solution of differential equation dydx=1x(3y2−1)\frac{d y}{d x}=\frac{1}{x\left(3 y^2-1\right)}, the value of constant term is_____ given that y=2\mathrm{y}=2 when x=1{x}=1.
View answer and explanation
Correct option: A
6
Option A (4) is correct because the area is calculated by integrating the function y = sqrt(36x) from x = 0 to x = 1.
234 Easy
234 of 494
Understanding +4 | -1
The order of the differential equation d2ydx2=1+(dydx)2\frac{{{d^2}y}}{{d{x^2}}} = \sqrt {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}} is
View answer and explanation
Correct option: B
2
Option B (2) is correct because the highest order derivative present in the differential equation is the second derivative.
235 Easy
235 of 494
Application +4 | -1
Consider the differential equation y2dx+(x−1y)dy=0y^2 d x+\left(x-\frac{1}{y}\right) d y=0. If y(1)=1y(1)=1, then xx is given by
View answer and explanation
Correct option: B
1+1y−e1/ye1+\frac{1}{y}-\frac{e^{1 / y}}{e}
Option B (1+1y−e1/ye1+\frac{{1}}{{y}}-\frac{{e^{1 / y}}}{{e}}) is correct because it is the solution to the given differential equation with the specified initial condition y(1)=1y(1)=1.
236 Easy
236 of 494
Application +4 | -1
The population p(t)p(t) at timet of a certain mouse species satisfies the differential equation
ddt(p(t))=0.5p(t)−450\frac{d}{d t}(p(t))=0.5 p(t)-450 If p(0)=850p(0)=850, then the time at which the population becomes zero is
View answer and explanation
Correct option: A
2ln⁡182 \ln 18
Option A (2ln⁡182 \ln 18) is correct because it represents the time calculated by solving the given differential equation with the initial condition and setting the population to zero.
237 Easy
237 of 494
Application +4 | -1
The general solution of the differential equation y(x2y+ex)dx−exdy=0y\left(x^2 y+e^x\right) d x-e^x d y=0, is
View answer and explanation
Correct option: B
x3y+3ex=3Cyx^3 y+3 e^x=3 C y
Option B (x3y+3ex=3Cyx^3 y+3 e^x=3 C y) is correct because it is the general solution obtained by integrating the given differential equation after rewriting it in an integrable form.
238 Easy
238 of 494
Application +4 | -1
In order to solve the differential equation
xcos⁡xdydx+y(xsin⁡x+cos⁡x)=1x \cos x \frac{d y}{d x}+y(x \sin x+\cos x)=1
the integrating factor is:
View answer and explanation
Correct option: B
xsec⁡xx \sec x
Option B, xsec⁡xx \sec x, is the correct integrating factor because it is derived from the exponential of the integral of P(x) after rewriting the differential equation in standard form.
239 Easy
239 of 494
Application +4 | -1
The solution of (1+xy)ydx+(1−xy)xdy=0(1+x y) y d x+(1-x y) x d y=0 is
View answer and explanation
Correct option: B
log⁡(xy)=1xy+k\log \left(\frac{x}{y}\right)=\frac{1}{x y}+k
Option B (log⁡(xy)=1xy+k\log \left(\frac{x}{y}\right)=\frac{1}{x y}+k) is the correct solution obtained by rearranging the equation, recognizing the exact differential, and integrating both sides.
240 Easy
240 of 494
Application +4 | -1
Solution of ℓn(dy/dx)=4x−2y−2,y(1)=1\ell \mathrm{n}(\mathrm{dy} / \mathrm{dx})=4 \mathrm{x}-2 \mathrm{y}-2, \mathrm{y}(1)=1, is -
View answer and explanation
Correct option: A
2e(2y+2)=e4x+e42 \mathrm{e}^{(2 y+2)}=\mathrm{e}^{4 x}+\mathrm{e}^4
Option A (2e(2y+2)=e4x+e42 \mathrm{e}^{(2 y+2)}=\mathrm{e}^{4 x}+\mathrm{e}^4) is correct because it represents the solution to the given differential equation with the initial condition y(1)=1y(1)=1.
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