The number of arbitrary constants in the particular solution of a differential equation of second order is (are):
- 0
- 1
- 2
- 3
Question types
DIFFERENTIAL EQUATIONS question types
761 questions across 7 question groups — pick any mix to generate a Maths paper with step-by-step answer keys.
761
Questions
7
Question groups
5
Question types
01
M.C.Q (1 Marks)
176 Q→02Assertion (A) & Reason (B) MCQ
6 Q→031 Marks
63 Q→042 Marks
35 Q→053 Marks
86 Q→064 Marks
385 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.
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Which of the following differential equations has $\text{y} = \text{c}_1\text{e}^\text{x} + \text{c}_2\text{e}^{-\text{x}}$ as the general solution?
View full solution →- $\frac{\text{d}^2\text{y}}{\text{dx}^2}+\text{y}=0$
- $\frac{\text{d}^2\text{y}}{\text{dx}^2}-\text{y}=0$
- $\frac{\text{d}^2\text{y}}{\text{dx}^2}+1=0$
- $\frac{\text{d}^2\text{y}}{\text{dx}^2}-1=0$
Which of the following differential equations has y = x as one of its particular solution?
View full solution →- $\frac{\text{d}^2\text{y}}{\text{dx}^2}-\text{x}^2\frac{\text{dy}}{\text{dx}}+\text{xy}=\text{x}$
- $\frac{\text{d}^2\text{y}}{\text{dx}^2}+\text{x}\frac{\text{dy}}{\text{dx}}+\text{xy}=\text{x}$
- $\frac{\text{d}^2\text{y}}{\text{dx}^2}-\text{x}^2\frac{\text{dy}}{\text{dx}}+\text{xy}=0$
- $\frac{\text{d}^2\text{y}}{\text{dx}^2}+\text{x}\frac{\text{dy}}{\text{dx}}+\text{xy}=0$
The degree of the differential equation
$\bigg(\frac{\text{d}^2\text{y}}{\text{dx}^2}\bigg)^3 + \bigg(\frac{\text{dy}}{\text{dx}}\bigg)^2+\text{sin} \bigg(\frac{\text{dy}}{\text{dx}}\bigg) + 1 =0 \ \text{is}$
View full solution →$\bigg(\frac{\text{d}^2\text{y}}{\text{dx}^2}\bigg)^3 + \bigg(\frac{\text{dy}}{\text{dx}}\bigg)^2+\text{sin} \bigg(\frac{\text{dy}}{\text{dx}}\bigg) + 1 =0 \ \text{is}$
- 3
- 2
- 1
- not defined.
Which of the following equation is a linear differential equation of order 3?
[Note: The original question asks for linear equation, but it should be linear differential equation]:
View full solution →[Note: The original question asks for linear equation, but it should be linear differential equation]:
- $\frac{\text{d}^3\text{y}}{\text{dx}^3}+\frac{\text{d}^2\text{y}\text{ dx}}{\text{dx}^2\text{dx}}+\text{y}=\text{x}$
- $\frac{\text{d}^3\text{y}}{\text{dx}^3}+\frac{\text{d}^2\text{y}}{\text{dx}^2}+\text{y}^2=\text{x}^2$
- $\text{x}\frac{\text{d}^3\text{y}}{\text{dx}^3}+\frac{\text{d}^2\text{y}}{\text{dx}^2}=\text{e}^\text{x}$
- $\frac{\text{d}^2\text{y}}{\text{dx}^2}+\frac{\text{dy}}{\text{dx}}=\log\text{x}$
Directions: In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices:
Assertion: If $\frac{\text{dy}}{\text{dy}}+\text{xy}=\text{x}^3\text{y}^3,\text{x}>0,\text{y}\geq0$ and $\text{y}(0)=1,$ then $\text{y}(1)=\frac{1}{\sqrt{2}}$
Reason: The differential equation is linear with integrating factor ex
View full solution →Assertion: If $\frac{\text{dy}}{\text{dy}}+\text{xy}=\text{x}^3\text{y}^3,\text{x}>0,\text{y}\geq0$ and $\text{y}(0)=1,$ then $\text{y}(1)=\frac{1}{\sqrt{2}}$
Reason: The differential equation is linear with integrating factor ex
- Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- Assertion is correct statement but Reason is wrong statement.
- Assertion is wrong statement but Reason is correct statement.
Directions: In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices:
Assertion: The elimination of four arbitrary constants in $\text{y}=(\text{c}_1+\text{c}_2+\text{c}_3\text{e}^\text{c}4)\text{x}$ results into a differential equation of the first order $\text{x}\frac{\text{dy}}{\text{dx}}=\text{y}$
Reason: Elimination of n arbitrary constants requires in general, a differential equation of the nth order.
View full solution →Assertion: The elimination of four arbitrary constants in $\text{y}=(\text{c}_1+\text{c}_2+\text{c}_3\text{e}^\text{c}4)\text{x}$ results into a differential equation of the first order $\text{x}\frac{\text{dy}}{\text{dx}}=\text{y}$
Reason: Elimination of n arbitrary constants requires in general, a differential equation of the nth order.
- Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- Assertion is correct statement but Reason is wrong statement.
- Assertion is wrong statement but Reason is correct statement.
Directions: In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices:
Assertion: $\text{x}\sin\text{x}\frac{\text{dy}}{\text{dx}}+(\text{x}+\text{x}\cos\text{x}+\sin \text{x}) \text{y}=\sin\text{xy},$
$(\frac{\pi}{2}) =1-\frac{2}{\pi}\Rightarrow \lim\limits_{\text{x}\rightarrow0}\text{y(x)}=\frac{1}{3}.$
Reason: The differential equation is linear with integrating factor $\text{x}(1-\cos\text{x})$
View full solution →Assertion: $\text{x}\sin\text{x}\frac{\text{dy}}{\text{dx}}+(\text{x}+\text{x}\cos\text{x}+\sin \text{x}) \text{y}=\sin\text{xy},$
$(\frac{\pi}{2}) =1-\frac{2}{\pi}\Rightarrow \lim\limits_{\text{x}\rightarrow0}\text{y(x)}=\frac{1}{3}.$
Reason: The differential equation is linear with integrating factor $\text{x}(1-\cos\text{x})$
- Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- Assertion is correct statement but Reason is wrong statement.
- Assertion is wrong statement but Reason is correct statement.
Directions: In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices:
Assertion: The differential equation of all circles in a plane must be of order 3.
Reason: If three points are non-collinear, then only one circle passes through these points.
Assertion: The differential equation of all circles in a plane must be of order 3.
Reason: If three points are non-collinear, then only one circle passes through these points.
- Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- Assertion is correct statement but Reason is wrong statement.
- Assertion is wrong statement but Reason is correct statement.
Directions: In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices:
Assertion: Order of the differential equation whose solution is $\text{y}=\text{c}_1\text{e}^{\text{x}+\text{c}_2}+\text{c}_3\text{e}^{\text{x}+\text{c}_4}$ is 4.
Reason: Order of the differential equation is equal to the number of independent arbitrary constants mentioned in the solution of the differential equation.
View full solution →Assertion: Order of the differential equation whose solution is $\text{y}=\text{c}_1\text{e}^{\text{x}+\text{c}_2}+\text{c}_3\text{e}^{\text{x}+\text{c}_4}$ is 4.
Reason: Order of the differential equation is equal to the number of independent arbitrary constants mentioned in the solution of the differential equation.
- Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- Assertion is correct statement but Reason is wrong statement.
- Assertion is wrong statement but Reason is correct statement.
Find the sum of the order and the degree of the following differential equation:
$\text{y = x} \bigg(\frac{\text{dy}}{\text{dx}}\bigg)^{3} + \frac{\text{d}^{2}{\text{y}}}{\text{d}}$
View full solution →$\text{y = x} \bigg(\frac{\text{dy}}{\text{dx}}\bigg)^{3} + \frac{\text{d}^{2}{\text{y}}}{\text{d}}$
Find the solution of the following differential equation:
$\text{x}\sqrt{(1 + \text{y}^{2})} \text{dx + y} \sqrt{( 1 + \text{x}^{2})} \text{dy} = 0$
View full solution →$\text{x}\sqrt{(1 + \text{y}^{2})} \text{dx + y} \sqrt{( 1 + \text{x}^{2})} \text{dy} = 0$
Find the sum of the order and the degree of the following differential equation:
$\text{y = x} \bigg(\frac{\text{dy}}{\text{dx}}\bigg)^{3} + \frac{\text{d}^{2}{\text{y}}}{\text{dx}^2}$
View full solution →$\text{y = x} \bigg(\frac{\text{dy}}{\text{dx}}\bigg)^{3} + \frac{\text{d}^{2}{\text{y}}}{\text{dx}^2}$
Write the differential equation representing the family of curves y = mx, where m is an arbitrary constant.
Find the differential equation representing the family of curves $\text{v} = \frac{\text{A}}{\text{r}} + \text{B},$ where A and B are arbitrary constants.
View full solution →$\text{Find} \frac{\text{dy}}{\text{dx}} \text{at } x = 1, \text{y} = \frac{\pi}{4} \text{if } { \sin}^{2}\text{y} + \cos x\text{y = K}.$
View full solution →Find the differential equation representing the family of curves y = aebx+5, where a and b are arbitrary constants.
Form the differential equation representing the family of curves y = e2x(a + bx), where ‘a’ and ‘b’ are arbitrary constants.
Determine the order and degree of the following differential equations. state also whether they are linear or non linear.
$\text{x}+\Big(\frac{\text{dy}}{\text{dx}}\Big)=\sqrt{1+\Big(\frac{\text{dy}}{\text{dx}}\Big)^2}$
View full solution →$\text{x}+\Big(\frac{\text{dy}}{\text{dx}}\Big)=\sqrt{1+\Big(\frac{\text{dy}}{\text{dx}}\Big)^2}$
Solve the following differential equations:
$\frac{\text{dy}}{\text{dx}}=(\cos^2\text{x}-\sin^2\text{x})\cos^2\text{y}$
View full solution →$\frac{\text{dy}}{\text{dx}}=(\cos^2\text{x}-\sin^2\text{x})\cos^2\text{y}$
Form the differential equation of the family of curves y = a cos(x + b), where a and b are arbitrary constants.
Solve the following differential equation:
View full solution →$4\frac{\text{dy}}{\text{dx}}+\text{8y}=\text{5e}^{-3x}$.
Solve the following differential equation:
View full solution →y(1 - x2) $\frac{\text{dy}}{\text{dx}}$ = x(1 + y2).
Verify that y = A cos x - b sin x is a solution of the differential equation.
View full solution →$\frac{\text{d}^{2} \text{y}}{\text{dx}^{2}}+\text{y}=0.$
Solve the following differential equation:
(y2 - x2) dy = 3xy dx.
$\text{If}\cos^{-1}\frac{x}{\text{a}} + \cos^{-1}\frac{y}{\text{b}} = \alpha, \text{Prove that}\frac{{x}^{2}}{\text{a}^{2}} - 2\frac{xy}{\text{ab}}\cos\alpha +\frac{{y}^{2}}{\text{b}^{2}} = \sin^{2}\alpha$
View full solution →Find the general solution of the differential equation
$\frac{\text{dy}}{\text{dx}}-\text{y}=\sin\text{x}.$
View full solution →$\frac{\text{dy}}{\text{dx}}-\text{y}=\sin\text{x}.$
Find the particular solution of the differential equation $\text{(x - y)} \frac{\text{dy}}{\text{dx}} = \text{(x + 2y),}$ given that y = 0 when x = 1.
View full solution →Find the equations of the tangent and normal to the curve$\frac{\text{x}^{2}}{\text{a}^{2}} - \frac{\text{y}^{2}}{\text{b}^{2}} = 1$ at the point ($\sqrt{2}$a, b).
View full solution →If y = P eax + Q ebx, show that
$\frac{\text{d}^2\text{y}}{\text{dx}^2}-(\text{a}+\text{b})\frac{\text{dy}}{\text{dx}}+\text{aby}=0$
View full solution →$\frac{\text{d}^2\text{y}}{\text{dx}^2}-(\text{a}+\text{b})\frac{\text{dy}}{\text{dx}}+\text{aby}=0$
A thermometer reading 80°F is taken outside. Five minutes later the thermometer reads 60°F. After another 5 minutes the thermometer reads 50°F. At any time t the thermometer reading be T°F and the outside temperature be S°F.
Based on the above information, answer the following questions.
View full solution →Based on the above information, answer the following questions.
- If $\lambda$ is posinve constant of propornonality, then $\frac{\text{dn}}{\text{dt}}$ is:
- $\lambda(\text{T - S})$
- $\lambda(\text{T + S})$
- $\lambda\text{T S}$
- $-\lambda(\text{T - S})$
- The value of T(S) is:
- 30°F
- 40°F
- 5D°F
- 60°F
- The value of T(10) is:
- 50°F
- 60°F
- 80°F
- 90°F
- Find the general solution of differential equation fanned in given situation.
- $\log\text{T}=\text{St + c}$
- $\log(\text{T}-\text{S})=\lambda{\text{t + c}}$
- $\log\text{T}=\text{tT + c}$
- $\log(\text{T + S})=\lambda{\text{t + c}}$
- Find the value of constant of integration c in the solution of differential equation formed in given situation.
- $\log(60\ -\ \text{S})$
- $\log(80\ +\ \text{S})$
- $\log(80\ -\ \text{S})$
- $\log(60\ +\ \text{S})$
A differential equation is said to be in the variable separable form if it is expressible in the form f(x) dx = g(y) dy.
The solution of this equation is given by
$\int\text{f(x)dx}=\int\text{g(y)dy}+\text{c},$ where c is the constant of integration.
Based on the above information, answer the following questions.
View full solution →The solution of this equation is given by
$\int\text{f(x)dx}=\int\text{g(y)dy}+\text{c},$ where c is the constant of integration.
Based on the above information, answer the following questions.
- If the solution of the differential equation $\frac{\text{dy}}{\text{dx}}=\frac{\text{ax+3}}{\text{2y+f}}$ represents a circle, then the value of 'a' is:
- 2
- -2
- 3
- -4
- The differential equation $\frac{\text{dy}}{\text{dx}}=\frac{\sqrt{1-\text{y}^2}}{\text{y}}$ determines a family of circle with.
- Variable radii and fixed centre (0, 1)
- Variable radii and fixed centre (0, -1)
- Fixed radius 1 and variable centre on x-axis
- Fixed radius 1 and variable centre on y-axis
- If = y'+ 1, y(0) = 1, then y (In 2) =
- 1
- 2
- 3
- 4
- The solution of the differential equation $\frac{\text{dy}}{\text{dx}}=\text{e}^\text{x-y}+\text{x}^2\text{e}^\text{-y}$ is:
- $\text{e}^\text{x}=\frac{\text{y}^3}{3}+\text{e}^\text{y}+\text{c}$
- $\text{e}^\text{y}=\frac{\text{x}^2}{3}+\text{e}^\text{x}+\text{c}$
- $\text{e}^\text{y}=\frac{\text{x}^3}{3}+\text{e}^\text{x}+\text{c}$
- None of these
- If $\frac{\text{dy}}{\text{dx}}=\text{y}\sin2\text{x},\ \text{y}(0)=1,$ then its solution is:
- $\text{y}=\text{e}^{\sin^2}\text{x}$
- $\text{y}={\sin^2}\text{x}$
- $\text{y}={\cos^2}\text{x}$
- $\text{y}=\text{e}^{\cos^2}\text{x}$
It is known that, if the interest is compounded continuously, the principal changes at the rate equal to the product of the rate of bank interest per annum and the principal. Let P denotes the principal at any time t and rate of interest be r% per annum.
Based on the above information, answer the following question.
View full solution →Based on the above information, answer the following question.
- Find the value of $\frac{\text{dP}}{\text{dt}}.$
- $\frac{\text{Pr}}{1000}$
- $\frac{\text{Pr}}{100}$
- $\frac{\text{Pr}}{10}$
- $\text{Pr}$
- If P0 be the initial principal, then find the solution of differential equation formed in given situation.
- $\log\Big(\frac{\text{P}}{\text{P}_0}\Big)=\frac{\text{rt}}{100}$
- $\log\Big(\frac{\text{P}}{\text{P}_0}\Big)=\frac{\text{rt}}{10}$
- $\log\Big(\frac{\text{P}}{\text{P}_0}\Big)=\text{rt}$
- $\log\Big(\frac{\text{P}}{\text{P}_0}\Big)=100\text{rt}$
- If the interest is compounded continuously at 5% per annum, in how many years will ₹ 100 double itself?
- 12.728 years
- 14.789 years
- 13.862 years
- 15.872 years
- At what interest rate will ₹ 100 double itself in 10 years? $(\log_\text{e}2 = 0.6931 ).$
- 9.66%
- 8.239%
- 7.341%
- 6.931%
- How much will ₹ 1000 be worth at 5% interest after 10 years? (e0.5 = 1.648).
- ₹ 1648
- ₹ 1500
- ₹ 1664
- ₹ 1572
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:
- $\frac{4-\pi^2}{2}$
- $\frac{8-\pi^2}{16}$
- $\frac{8-\pi^2}{4}$
- $\frac{4+\pi^2}{2}$
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