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JEE • MATHEMATICS • DIFFERENTIAL EQUATIONS

Differential equations practice · Mixed difficulty

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181 Easy
181 of 975
Application +4 | -1
The differential equation of the family of parabolas having vertex at the origin and the yy-axis as the axis of symmetry, is
View answer and explanation
Correct option: C
xy1−2y=0x y_1-2 y=0
Option C (xy1−2y=0x y_1-2 y=0) is correct because it represents the differential equation of the given family of parabolas.
182 Easy
182 of 975
Application +4 | -1
The general solution of y′+sin⁡x+y2=sin⁡x−y2y^{\prime}+\sin \frac{x+y}{2}=\sin \frac{x-y}{2} is
View answer and explanation
Correct option: C
log⁡∣tan⁡(y/4)∣=C−2sin⁡x/2\log |\tan (y / 4)|=C-2 \sin x / 2
Option C (log⁡∣tan⁡(y/4)∣=C−2sin⁡x/2\log |\tan (y / 4)|=C-2 \sin x / 2) is correct because it is the general solution of the given differential equation.
183 Easy
183 of 975
Understanding +4 | -1
The differential equation dydx+Py=Qyn\frac{{dy}}{{dx}} + Py = Q{y^n} , n>2n > 2 can be reduced to linear form by substituting
View answer and explanation
Correct option: D
z=y1−nz=y^{1-n}
Option D (z=y1−nz=y^{1-n}) is the correct substitution to transform the Bernoulli differential equation into a linear form.
184 Easy
184 of 975
Application +4 | -1
Solution of the equation dydx+y(x+y)x2=0\frac{d y}{d x}+\frac{y(x+y)}{x^2}=0 is -
View answer and explanation
Correct option: C
x2y=c(2x+y)x^2 y=c(2 x+y)
Option A (x2y=c(2x+y)x^2 y=c(2 x+y)) is correct because solving the differential equation using Bernoulli's equation leads to this solution.
185 Easy
185 of 975
Understanding +4 | -1
Let y=[acos⁡x+(b+c)sin⁡x]ex+dy=[a \cos x+(b+c) \sin x] e^{x+d} where a,b,c,da, b, c, d are parameters. The order of the differential equation formed by eliminating of parameters is
View answer and explanation
Correct option: A
2
Option A (2) is correct because the given function can be simplified to contain only two independent parameters after accounting for the constants.
186 Easy
186 of 975
Application +4 | -1
The solution of the differential equation dydx=1+y21+x2\frac{d y}{d x}=\frac{1+y^2}{1+x^2}, is
View answer and explanation
Correct option: B
dy−x=C(1+xy)d y-x=C(1+x y)
Option B (dy−x=C(1+xy)dy - x = C(1 + xy)) is correct because it is the result of separating variables, integrating, and using the tangent subtraction formula.
187 Easy
187 of 975
Application +4 | -1
The general solution of the differential equation dydx=ey+x+ey−x\frac{d y}{d x}=e^{y+x}+e^{y-x} is
View answer and explanation
Correct option: B
e−y=e−x−ex+Ce^{-y}=e^{-x}-e^x+C
Option B (e−y=e−x−ex+Ce^{-y}=e^{-x}-e^x+C) is correct because it is the solution obtained by separating variables and integrating.
188 Medium
188 of 975
Application +4 | -1
Solution of dydx=x(2log⁡x+1)sin⁡y+ycos⁡y\frac{d y}{d x}=\frac{x(2 \log x+1)}{\sin y+y \cos y} when y=πy=\pi at x=ex=e is
View answer and explanation
Correct option: B
ysin⁡y=x2log⁡x−e2y \sin y=x^2 \log x-e^2
Option B (ysin⁡y=x2log⁡x−e2y \sin y=x^2 \log x-e^2) is correct because it is the solution to the given differential equation with the specified initial condition.
189 Medium
189 of 975
Application +4 | -1
The general solution of the differential equation dydx+sin⁡(x+y2)=sin⁡(x−y2)\frac{{dy}}{{dx}} + \sin \left( {\frac{{x + y}}{2}} \right) = \sin \left( {\frac{{x - y}}{2}} \right) is
View answer and explanation
Correct option: B
log⁡tan⁡(y4)=c−2sin⁡(x2)\log \tan \left( {\frac{y}{4}} \right) = c - 2\sin \left( {\frac{x}{2}} \right)
Option B (log⁡tan⁡(y4)=c−2sin⁡(x2)\log \tan \left( {\frac{y}{4}} \right) = c - 2\sin \left( {\frac{x}{2}} \right)) is correct because it is the general solution of the given differential equation after integration and simplification.
190 Medium
190 of 975
Application +4 | -1
The general solution of dydx=2x−yx+2y\frac{{dy}}{{dx}} = \frac{{2x - y}}{{x + 2y}} is
View answer and explanation
Correct option: B
x2−xy−y2=c{x^2} - xy - {y^2} = c
Option B (x2−xy−y2=c{x^2} - xy - {y^2} = c) is the correct general solution obtained by solving the given homogeneous differential equation using the substitution y=vxy = vx.
191 Easy
191 of 975
Application +4 | -1
If x4+y4=x2y2x^4+y^4=x^2 y^2 then dydx\frac{d y}{d x} is given by
View answer and explanation
Correct option: C
xy⋅y2−2x22y2−x2\frac{x}{y} \cdot \frac{y^2-2 x^2}{2 y^2-x^2}
Option C (xy⋅y2−2x22y2−x2\frac{x}{y} \cdot \frac{y^2-2 x^2}{2 y^2-x^2}) is correct because it is the result of implicit differentiation and algebraic manipulation.
192 Easy
192 of 975
Application +4 | -1
The solution of y′(y−x−4)=x+y−2y^{\prime}(y-x-4)=x+y-2 is given by
View answer and explanation
Correct option: B
x2+2xy−y2−4x+8y=Cx^2+2 x y-y^2-4 x+8 y=C
Option B is correct because the given differential equation can be solved by transforming it into a homogeneous equation and then integrating, resulting in x2+2xy-y2-4x+8y=C.
193 Easy
193 of 975
Application +4 | -1
The solution to the differntial equation ydx+xlog⁡(yx)dy−2xdy=0y d x+x \log \left(\frac{y}{x}\right) d y-2 x d y=0 is
View answer and explanation
Correct option: B
cy=log⁡∣yx∣−1c y=\log \left|\frac{y}{x}\right|-1
Option B (cy=log⁡∣yx∣−1c y=\log \left|\frac{y}{x}\right|-1) is correct because it is the solution to the given homogeneous differential equation.
194 Medium
194 of 975
Application +4 | -1
The degree and order of the differential equation y=px+a2p2+b23y = px + \sqrt[3]{{{a^2}{p^2} + {b^2}}}, where p=dydxp = \frac{{dy}}{{dx}}, are respectively
View answer and explanation
Correct option: A
3,13,1
Option A (3,1) is correct because after cubing the equation to remove the cube root, the highest order derivative is dy/dx (order 1), and its power is 3 (degree 3).
195 Medium
195 of 975
Understanding +4 | -1
The order of differential equation whose general solution is given by y=c1e2x+c2+c3ex+c4sin⁡(x+c5)y = {c_1}{e^{2x + {c_2}}} + {c_3}{e^x} + {c_4}\sin \left( {x + {c_5}} \right) is
View answer and explanation
Correct option: B
4
Option B (4) is correct because the general solution can be rewritten with 4 independent arbitrary constants: y=Ae2x+c3ex+Bsin⁡x+Dcos⁡xy = A{e^{2x}} + {c_3}{e^x} + B\sin x + D\cos x, thus the order of the differential equation is 4.

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196 Medium
196 of 975
Application +4 | -1
If the solution y=y(x)y=y(x) of the differential equation (x4+2x3+3x2+2x+2)dy−(2x2+2x+3)dx=0\left(x^4+2 x^3+3 x^2+2 x+2\right) d y-\left(2 x^2+2 x+3\right) d x=0 satisfies y(−1)=−π4y(-1)=-\frac{\pi}{4}, then y(0)y(0) is equal to:
View answer and explanation
Correct option: C
π4\frac{\pi}{4}
Option C (π4\frac{\pi}{4}) is correct because, after solving the differential equation and applying the given condition, y(0) equals π4\frac{\pi}{4}.
197 Medium
197 of 975
Application +4 | -1
Solution of the differential equation dydx+ysec⁡2x=tan⁡xsec⁡2x\frac{{dy}}{{dx}} + y{\sec ^2}x = \tan x{\sec ^2}x is
View answer and explanation
Correct option: A
y=tan⁡x−1+ce−tan⁡xy = \tan x - 1 + c{e^{ - \tan x}}
Option A (y=tan⁡x−1+ce−tan⁡xy = \tan x - 1 + c{e^{ - \tan x}}) is correct because it is the solution to the given first-order linear differential equation, obtained by finding the integrating factor and integrating.
198 Easy
198 of 975
Understanding +4 | -1
Assertion : The degree of the differential equation

d3ydx3+2(d2ydx2)32+2y=0 is zero \frac{d^3 y}{d x^3}+2\left(\frac{d^2 y}{d x^2}\right)^{\frac{3}{2}}+2 y=0 \text { is zero }

Reason: The degree of a differential equation is not defined if it is not a polynomial eq in its derivatives.
View answer and explanation
Correct option: D
Assertion is incorrect, Reason is correct.
Assertion is incorrect, Reason is correct because the degree of a differential equation is only defined if it's a polynomial equation in its derivatives, and the given equation is not.
199 Easy
199 of 975
Application +4 | -1
The degree of the differential equation dydx+x=\frac{d y}{d x}+x= (y+xdydx)−4\left(y+x \frac{d y}{d x}\right)^{-4} is
View answer and explanation
Correct option: D
5
Option D (5) is correct because the degree of a differential equation is the highest power of the highest order derivative after the equation has been rationalized.
200 Medium
200 of 975
Application +4 | -1
The solution of the differential equation dydx=1xy[x2sin⁡y2+1]\frac{d y}{d x}=\frac{1}{x y\left[x^2 \sin y^2+1\right]} is
View answer and explanation
Correct option: A
x2(cos⁡y2−sin⁡y2−2ce−y2)=2x^2\left(\cos y^2-\sin y^2-2 c e^{-y^2}\right)=2
Option A (x2(cos⁡y2−sin⁡y2−2ce−y2)=2x^2(\cos y^2 - \sin y^2 - 2ce^{-y^2}) = 2) is correct because it is the solution obtained by rewriting the equation, using a substitution, and solving the resulting linear differential equation.
201 Easy
201 of 975
Analyzing +4 | -1
Let f(x)f(x) be differentiable on the interval (0,∞)(0, \infty) such that f(1)=1f(1)=1, and lim⁡t→xt2f(x)−x2f(t)t−x=1\lim\limits _{t \rightarrow x} \frac{t^2 f(x)-x^2 f(t)}{t-x}=1 for each x>0x>0. Then f(x)f(x) is
View answer and explanation
Correct option: A
13x+23x2\frac{1}{3 x}+\frac{2}{3} x^2
Option A is correct because it satisfies the given limit condition and the initial condition f(1)=1 after solving the differential equation.
202 Medium
202 of 975
Application +4 | -1
The solution of the differential equation (x2−yx2)dydx+y2+xy2=0\left( {{x^2} - y{x^2}} \right)\frac{{dy}}{{dx}} + {y^2} + x{y^2} = 0 is
View answer and explanation
Correct option: A
log⁡(xy)=1x+1y+c\log \left( {\frac{x}{y}} \right) = \frac{1}{x} + \frac{1}{y} + c
Option A (log⁡(xy)=1x+1y+c\log \left( {\frac{x}{y}} \right) = \frac{1}{x} + \frac{1}{y} + c) is correct because it represents the solution obtained by separating variables and integrating the given differential equation.
203 Medium
203 of 975
Application +4 | -1
The solution of the differential equation (2y−1)dx−(2x+3)dy=0\left( {2y - 1} \right)dx - \left( {2x + 3} \right)dy = 0, is
View answer and explanation
Correct option: B
2x+32y−1=C\frac{{2x + 3}}{{2y - 1}} = C
Option B (2x+32y−1=C\frac{{2x + 3}}{{2y - 1}} = C) is correct because it is the general solution obtained by separating variables and integrating the given differential equation.
204 Easy
204 of 975
Application +4 | -1
The solution of dydx−y=1,y(0)=1\frac{{dy}}{{dx}} - y = 1,y\left( 0 \right) = 1 is given by y(x)=y\left( x \right) =
View answer and explanation
Correct option: D
2exp⁡(x)−12\exp \left( x \right) - 1
Option D (2exp⁡(x)−12\exp \left( x \right) - 1) is the correct solution obtained by solving the given differential equation with the initial condition y(0) = 1.
205 Medium
205 of 975
Application +4 | -1
The general solution of the differential equation (x−y2)dx+y(5x+y2)dy=0\left(x-y^2\right) d x+y\left(5 x+y^2\right) d y=0 is :
View answer and explanation
Correct option: A
(y2+x)4=C∣(y2+2x)3∣\left(y^2+x\right)^4=C\left|\left(y^2+2 x\right)^3\right|
Option B ((y2+2x)4=C∣(y2+x)3∣\left(y^2+2 x\right)^4=C\left|\left(y^2+x\right)^3\right|) is incorrect. The correct answer is A. This option has the terms switched in the final equation.
206 Easy
206 of 975
Application +4 | -1
The solution of ydx−xdy+3x2y2ex3dx=0y d x-x d y+3 x^2 y^2 e^{x^3} d x=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 the correct solution obtained by rearranging the equation and integrating both sides after recognizing the exact differential.
207 Medium
207 of 975
Application +4 | -1
If a curve passing through (1,1)(1,1) satisfies the differential equation x2dy=y2dx+xyd(xy)x^2 d y=y^2 d x+x y d(x y), then the equation of the curve is
View answer and explanation
Correct option: B
log⁡∣xy∣=1x−1y\log |x y|=\frac{1}{x}-\frac{1}{y}
Option B (log⁡∣xy∣=1x−1y\log |x y|=\frac{1}{x}-\frac{1}{y}) is correct because it represents the solution to the given differential equation after integration and applying the initial condition.
208 Medium
208 of 975
Application +4 | -1
General solution of the differential equation (xlog⁡x)dydx+y=2xlog⁡x(x \log x) \frac{d y}{d x}+y=\frac{2}{x} \log x is
View answer and explanation
Correct option: D
xylog⁡x=cx−2(1+log⁡∣x∣)x y \log x=c x-2(1+\log |x|)
Option D (xylog⁡x=cx−2(1+log⁡∣x∣)x y \log x=c x-2(1+\log |x|)) is correct because it is the solution obtained after solving the given linear differential equation.
209 Medium
209 of 975
Application +4 | -1
The solution of the differential equation ydx+xdyydx−xdy=x2exyy4\frac{y d x+x d y}{y d x-x d y}=\frac{x^2 e^{x y}}{y^4} satisfying y(0)=1y(0)=1, is
View answer and explanation
Correct option: B
x3=3y3(1−e−xy)x^3=3 y^3\left(1-e^{-x y}\right)
Option B (x3=3y3(1−e−xy)x^3=3 y^3(1-e^{-x y})) is correct because it is the solution to the differential equation that satisfies the initial condition y(0)=1.
210 Easy
210 of 975
Application +4 | -1
Let the solution curve y=y(x)y=y(x) of the differential equation dydx−3x5tan⁡−1(x3)(1+x6)3/2y=2xexp⁡{x3−tan⁡−1x3(1+x6)}\frac{d y}{d x}-\frac{3 x^5 \tan ^{-1}\left(x^3\right)}{\left(1+x^6\right)^{3 / 2}} y=2 x \exp \left\{\frac{x^3-\tan ^{-1} x^3}{\sqrt{\left(1+x^6\right)}}\right\} pass through the origin. Then y(1)y(1) is equal to
View answer and explanation
Correct option: B
exp⁡(4−π42)\exp \left(\frac{4-\pi}{4 \sqrt{2}}\right)
Option B (exp((4-π)/(4√2))) is correct because it is the value of y(1) obtained after solving the given differential equation and applying the initial condition y(0)=0.
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