The integrating factor of the differential equation $\left(x+2 y^2\right) \frac{d y}{d x}=y(y>0)$ is :
- A$\frac{1}{x}$
- B$x$
- C$y$
- D$\frac{1}{y}$
Question types
Differential Equations question types
198 questions across 7 question groups — pick any mix to generate a Maths paper with step-by-step answer keys.
198
Questions
7
Question groups
5
Question types
01
M.C.Q (1 Marks)
59 Q→02Assertion (A) & Reason (B) MCQ
7 Q→031 Marks
28 Q→042 Marks
35 Q→053 Marks
20 Q→064 Marks
39 Q→07Case study (4 Marks)
10 Q→Sample Questions
Differential Equations questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
The order of the following differential equation $\frac{d^3 y}{d x^3}+x\left(\frac{d y}{d x}\right)^5=4 \log \left(\frac{d^4 y}{d x^4}\right)$ is:
- Anot defined
- B3
- ✓4
- D5
Answer: C.
View full solution →The degree of the differential equation $\left(y^{\prime \prime}\right)^2+\left(y^{\prime}\right)^3$ $=x \sin \left(y^{\prime}\right)$ is :
- A1
- B2
- C3
- Dnot defined
The general solution of the differential equation $x d y+y d x=0$ is:
- A$x y=c$
- B$x+y=c$
- C$x^2+y^2=c^2$
- D$\log y=\log x+c$
The degree and order of differential equation $y^{\prime \prime 2}+\log \left(y^{\prime}\right)=x^5$ respectively are:
- Anot defined, 5
- Bnot defined, 2
- C5, not defined
- D2,2
Assertion (A) : If $\frac{d y}{d x}+x y=x^3 y^3, x>0, y \geq 0$ and $y(0)=1$, then $y(1)=\frac{1}{\sqrt{2}}$.
Reason (R) : The differential equation is linear with integrating factor $e^x$.
Reason (R) : The differential equation is linear with integrating factor $e^x$.
- ABoth (A) and (R) are true and (R) is the correct explanation of (A).
- BBoth (A) and (R) are true but (R) is not the correct explanation of (A).
- C(A) is true but (R) is false.
- D(A) is false but (R) is true.
Assertion (A) : $x \sin x \frac{d y}{d x}+(x+x \cos x+\sin x)$ $y=\sin x, y\left(\frac{\pi}{2}\right)=1-\frac{2}{\pi} \Rightarrow y=\frac{x-\sin x}{x(1-\cos x)}$
Reason (R) : The differential equation is linear with integrating factor $x(1-\cos x)$.
Reason (R) : The differential equation is linear with integrating factor $x(1-\cos x)$.
- ABoth (A) and (R) are true and (R) is the correct explanation of (A).
- BBoth (A) and (R) are true but (R) is not the correct explanation of (A).
- C(A) is true but (R) is false.
- D(A) is false but (R) is true.
Assertion (A) : $y=a \sin x+b \cos x$ is a general solution of $y^{\prime \prime}+y=0$.
Reason (R): $y=a \sin x+b \cos x$ is a trigonometric function.
Reason (R): $y=a \sin x+b \cos x$ is a trigonometric function.
- ABoth (A) and (R) are true and (R) is the correct explanation of (A).
- ✓Both (A) and (R) are true but (R) is not the correct explanation of (A).
- C(A) is true but (R) is false.
- D(A) is false but (R) is true.
Answer: B.
View full solution →Assertion (A) : A differential equation of the form $y f(x y) d x+x g(x y) d y=0$ can be converted into homogeneous differential equation by substituting $x y=t$
Reason (R) : A differential equation is called homogeneous if $f(\lambda x, \lambda y)=\lambda^0 f(x, y)$.
Reason (R) : A differential equation is called homogeneous if $f(\lambda x, \lambda y)=\lambda^0 f(x, y)$.
- ABoth (A) and (R) are true and (R) is the correct explanation of (A).
- BBoth (A) and (R) are true but (R) is not the correct explanation of (A).
- C(A) is true but (R) is false.
- ✓(A) is false but (R) is true.
Answer: D.
View full solution →Assertion (A) : Order of the differential equation whose solution is $y=c_1 e^{x+c_2}+c_3 e^{x+c_4}$ is 4.
Reason (R) : Order of the differential equation is equal to the number of independent arbitrary constants mentioned in the solution of the differential equation.
Reason (R) : Order of the differential equation is equal to the number of independent arbitrary constants mentioned in the solution of the differential equation.
- ABoth (A) and (R) are true and (R) is the correct explanation of (A).
- BBoth (A) and (R) are true but (R) is not the correct explanation of (A).
- C(A) is true but (R) is false.
- ✓(A) is false but (R) is true.
Answer: D.
View full solution →The general solution of the differential equation $\frac{y d x-x d y}{y}=0$ is
View full solution →The general solution of the differential equation ex dy + (y ex + 2x) dx = 0 is
The general solution of a differential equation of the type $\frac{d x}{d y}+\mathrm{P}_{1} x=\mathrm{Q}_{1}$ is
View full solution →Find the order and degree (if defined) of the differential equation $\frac{d^{4} y}{d x^{4}}-\sin \left(\frac{d^{3} y}{d x^{3}}\right)=0$
View full solution →Find the order and degree (if defined) of the differential equation $\left(\frac{d y}{d x}\right)^{3}-4\left(\frac{d y}{d x}\right)^{2}+7 y=\sin x$
View full solution →Solve the differential equation $y e^{\frac{x}{y}} d x=\left(x e^{\frac{x}{y}}+y^{2}\right) d y(y \neq 0)$
View full solution →Find the particular solution of the differential equation (1 + e2x)dy + (1 + y2)ex dx = 0, given that y = 1, when x = 0.
Find the equation of the curve passing through the point $\left( {0,\frac{\pi }{4}} \right)$whose differential equation is $\sin x\cos ydx + \cos x\sin ydy = 0$
View full solution →Find the general solution of the differential equation $\frac{d y}{d x}+\sqrt{\frac{1-y^{2}}{1-x^{2}}}=0$
View full solution →Verify that the function $x^{2}=2 y^{2} \log y$ (implicit or explicit) is a solution of the differential equation $\left(x^{2}+y^{2}\right) \frac{d y}{d x}-x y=0$
View full solution →Find a particular solution of the differential equation $(x+1) \frac{d y}{d x}=2 e^{-y}-1$ given that y = 0 when x = 0.
View full solution →Find the general solution of $\frac{{dy}}{{dx}} + \left( {\sec x} \right)y = \tan x\left( {0 \leq x < \frac{\pi }{2}} \right)$
View full solution →Find the general solution of $\frac{d y}{d x}+\frac{y}{x}=x^{2}$
View full solution →Find the general solution of $\frac{d y}{d x}+3 y=e^{-2 x}$
View full solution →Find the particular solution of the differential equation $\frac { d y } { d x } - 3 y \cot x = \sin 2 x$ , given that y = 2 when $x = \frac { \pi } { 2 }$.
View full solution →Find a particular solution of the differential equation (x - y) (dx + dy) = dx - dy, given that y = -1, when x = 0. (Hint: put x - y = t)
Prove that x2 - y2 = c(x2 + y2)2 is the general solution of differential equation $(x^3-3xy^2)dx=(y^3-3x^2y)dy$, where c is a parameter.
View full solution →Find a particular solution of the differential equation $ \frac { d y } { d x } + y \cot x = 4 x \; cosec \; x$, x $\neq$ 0 given that y = 0, when $ x = \frac { \pi } { 2 }$.
View full solution →Solve the differential equation $\left( {\frac{{{e^{ - 2\sqrt x }}}}{{\sqrt x }} - \frac{y}{{\sqrt x }}} \right)\frac{{dx}}{{dy}}$ = 1 (x $\neq$ 0)
View full solution →Find the general solution $ x \frac { d y } { d x } + y - x + x y \cot x = 0 \ (x \neq 0)$.
View full solution →Order: The order of a differential equation is the order of the highest order derivative appearing in the differential equation.
Degree: The degree of differential equation is the power of the highest order derivative, when differential coefficients are made free from radicals and fractions. Also, differential equation must be a polynomial equation in derivatives for the degree to be defined.
Based on the above information, answer the following questions.
View full solution →Degree: The degree of differential equation is the power of the highest order derivative, when differential coefficients are made free from radicals and fractions. Also, differential equation must be a polynomial equation in derivatives for the degree to be defined.
Based on the above information, answer the following questions.
- Find the degree of the differential equation $2\frac{\text{d}^2\text{y}}{\text{dx}^2}+3\sqrt{1-\Big(\frac{\text{dy}}{\text{dx}}\Big)^2-\text{y}=0}.$
- 3
- 4
- 3
- 1
- Order and degree of the differential equation $\text{y}\frac{\text{dy}}{\text{dx}}=\frac{\text{x}}{\frac{\text{dy}}{\text{dx}}+\Big(\frac{\text{dy}}{\text{dx}}\Big)^3}$ are respectively.
- 1, 1
- 1, 2
- 1, 3
- 1, 4
- Find order and degree of the equation y'" + y2 + ey' = 0.
- Order = 3, degree = undefined.
- Order = 1, degree = 3.
- Order = 2, degree = undefined.
- Order = 1, degree = 2.
- Determine degree of the differential equation $(\sqrt{\text{a+x}})\times\Big(\frac{\text{dy}}{\text{dx}}\Big)+\text{x}=0.$
- 3
- Not defined
- 1
- 2
- Order and degree of the differential equation $\Bigg(1+\Big(\frac{\text{dy}}{\text{dx}}\Big)^3\Bigg)^\frac{7}{3}=7\frac{\text{d}^2\text{y}}{\text{dx}^2}$ are respectively.
- 2, 1
- 2, 3
- 1, 3
- $1,\ \frac{7}{3}$
If an equation is of the form $\frac{\text{dy}}{\text{dx}}+\text{Py}=\text{Q},$ where P, Qare functions of x, then such equation is known as linear differential equation. Its solution is given by
$\text{y}\times\text{(I.F.)}=\int\text{Q}\times\text{(I.F.)}\text{dx}+\text{c},$ where $\text{I.F.}=\text{e}^{\int\text{pdx}}.$
Now, suppose the given equation is $(1+\sin\text{x})\frac{\text{dy}}{\text{dx}}+\text{y}\cos\text{x}+\text{x}=0.$
Based on the above information, answer the following questions.
View full solution →$\text{y}\times\text{(I.F.)}=\int\text{Q}\times\text{(I.F.)}\text{dx}+\text{c},$ where $\text{I.F.}=\text{e}^{\int\text{pdx}}.$
Now, suppose the given equation is $(1+\sin\text{x})\frac{\text{dy}}{\text{dx}}+\text{y}\cos\text{x}+\text{x}=0.$
Based on the above information, answer the following questions.
- The value of P and Q respectively are:
- $\frac{\sin\text{x}}{1+\cos\text{x}},\ \frac{\text{x}}{1+\sin\text{x}}$
- $\frac{\cos\text{x}}{1+\sin\text{x}},\ \frac{\text{-x}}{1+\sin\text{x}}$
- $\frac{-\cos\text{x}}{1+\sin\text{x}},\ \frac{\text{x}}{1+\sin\text{x}}$
- $\frac{\cos\text{x}}{1+\sin\text{x}},\ \frac{\text{x}}{1+\sin\text{x}}$
- The value of I.F is:
- $1-\sin\text{x}$
- $\cos\text{x}$
- $1+\sin\text{x}$
- $1-\cos\text{x}$
- Solution of given equation is:
- $\text{y}(1-\sin\text{x})=\text{x+c}$
- $\text{y}(1+\sin\text{x})=-\text{x}^2+\text{c}$
- $\text{y}(1-\sin\text{x})=\frac{\text{-x}^2}{2}\text{+c}$
- $\text{y}(1+\sin\text{x})=\frac{\text{-x}^2}{2}\text{+c}$
- If y(0) = 1, then y equals
- $\frac{2-\text{x}^2}{2(1+\sin\text{x})}$
- $\frac{2+\text{x}^2}{2(1+\sin\text{x})}$
- $\frac{2-\text{x}^2}{2(1-\sin\text{x})}$
- $\frac{2+\text{x}^2}{2(1-\sin\text{x})}$
- Value of is $\text{y}\Big(\frac{\pi}{2}\Big)$ is:
- $\frac{4-\pi^2}{2}$
- $\frac{8-\pi^2}{16}$
- $\frac{8-\pi^2}{4}$
- $\frac{4+\pi^2}{2}$
If the equation is of the form $\frac{\text{dy}}{\text{dx}}=\text{py}=\text{Q},$ where P, Q are functions of x, then the solution of the differential equation is given by $\text{ye}^{\int\text{pdx}}=\int\text{Q e}^{\int\text{pdx}}\text{dx}+\text{c},$ where $\text{e}^{\int\text{pdx}}$ is called the integrating factor (I.F.).
Based on the above information, answer the following questions.
View full solution →Based on the above information, answer the following questions.
- The integrating factor of the differential equation $\sin\text{x}\frac{\text{dy}}{\text{dx}}+2\text{y}\cos\text{x}=1$ is $(\sin\text{x})^\lambda,$ where $\lambda=$
- 0
- 1
- 2
- 3
- Integrating factor of the differential equation $(1-\text{x}^2)\frac{\text{dy}}{\text{dx}}-\text{xy}=1$ is:
- $-\text{x}$
- $\frac{\text{x}}{1+\text{x}^2}$
- $\sqrt{1-\text{x}^2}$
- $\frac{1}{2}\log(1-\text{x}^2)$
- The solution of $\frac{\text{dy}}{\text{dx}}+\text{y}=\text{e}^{-\text{x}},\text{ y}(0)=0,$ is:
- $\text{y}=\text{e}^\text{x}(\text{x}-1)$
- $\text{y}=\text{xe}^{-\text{x}}$
- $\text{y}=\text{xe}^{-\text{x}}+1$
- $\text{y}=(\text{x}+1)\text{e}^{-\text{x}}$
- General solution of $\frac{\text{dy}}{\text{dx}}+\text{y}\tan\text{x}=\sec\text{x}$ is:
- $\text{y}\sec\text{y}=\tan\text{x}+\text{c}$
- $\text{y}\tan\text{x}=\sec\text{x}+\text{c}$
- $\tan\text{x}=\text{y}\tan\text{x}+\text{c}$
- $\text{x}\sec\text{x}=\tan\text{y}+\text{c}$
- The integrating factor of differential equation $\frac{\text{dy}}{\text{dx}}-3\text{y}=\sin2\text{x}$ is:
- $\text{e}^{3\text{x}}$
- $\text{e}^{-2\text{x}}$
- $\text{e}^{-3\text{x}}$
- $\text{xe}^{-3\text{x}}$
In a college hostel accommodating 1000 students, one of the hostellers came in carrying Corona virus, and the hostel was isolated. The rate at which the virus spreads is assumed to be proportional to the product of the number of infected students and remaining students. There are 50 infected students after 4 days.
Based on the above information, answer the following questions.
View full solution →Based on the above information, answer the following questions.
- If n(I) denote the number of students infected by Corona virus at any time I, then maximum value of n(I) is:
- 50
- 100
- 500
- 1000
- $\frac{\text{dn}}{\text{dt}}$ is proporuona to:
- n(1000 - n)
- n(100 + n)
- n(100 - n)
- n(100 + n)
- The value of n(4) is:
- 1
- 50
- 100
- 1000
- The most general solution of differential equation formed in given situation is:
- $\frac{1}{1000}\log\Big(\frac{1000-\text{n}}{\text{n}}\Big)=\lambda\text{t}+\text{c}$
- $\log\Big(\frac{\text{n}}{100-\text{n}}\Big)=\lambda\text{t}+\text{c}$
- $\frac{1}{1000}\log\Big(\frac{\text{n}}{1000-\text{n}}\Big)=\lambda\text{t}+\text{c}$
- None of these.
- The value of n at any time is given by:
- $\text{n(t)}=\frac{1000}{1+999\text{e}^{-0.9906\text{t}}}$
- $\text{n(t)}=\frac{1000}{1-999\text{e}^{-0.9906\text{t}}}$
- $\text{n(t)}=\frac{100}{1-999\text{e}^{-0.9906\text{t}}}$
- $\text{n(t)}=\frac{100}{1+999\text{e}^{-0.9906\text{t}}}$
In a murder investigation, a corpse was found by a detective at exactly 8 p.m. Being alert, the detective measured the body temperature and found it to be 70ºF Two hours later, the detective measured the body temperature again and found it to be 60ºF, where the room temperature is 50ºF Also, it is given the body temperature at the time of death was normal, i.e., 98.6ºF.
Let T be the temperature of the body at any time t and initial time is taken to be 8 p.m.
Based on the above information, answer the following questions.
View full solution →Let T be the temperature of the body at any time t and initial time is taken to be 8 p.m.
Based on the above information, answer the following questions.
- By Newton's law of cooling, $\frac{\text{dT}}{\text{dt}}$ is proportional to:
- T - 60
- T - 50
- T - 70
- T - 98.6
- When t = 0, then body temperature is equal to:
- 50ºF
- 60ºF
- 70ºF
- 98.6ºF
- When t = 2, then body temperature is equal to:
- 50ºF
- 60ºF
- 70ºF
- 98.6ºF
- The value of T at any time t is:
- $50+20\Big(\frac{1}{2}\Big)^\text{t}$
- $50+20\Big(\frac{1}{2}\Big)^\text{t-1}$
- $50+20\Big(\frac{1}{2}\Big)^\frac{\text{t}}{2}$
- None of these
- If it is given that $\log_\text{e} (2.43) = 0.88789$ and $\log_\text{e} (0.5) = -0.69315,$ then the time at which the murder occur is:
- 7:30 p.m.
- 5:30 p.m.
- 6:00 p.m.
- 5:00 p.m.
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