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All questions of Differential equation for Mathematics Exam

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If f is twice differentiable function such that f ''(x) = - f(x), f '(x) = g(x) and h(x) = [f(x)]2 + [g(x)]2, also if h(5) = 11, then h(10) is equal to
  • a)
    22
  • b)
    121
  • c)
    16
  • d)
    None of these
Correct answer is option 'D'. Can you explain this answer?

Qiana Sharma answered
Given Information

A function f(x) is twice differentiable and satisfies the following conditions:


  • f(x) = -f(x)

  • f(x) = g(x)

  • h(x) = [f(x)]^2 * [g(x)]^2

  • h(5) = 11



Solution

To find the value of h(10), we need to use the given information and properties of the functions f(x), g(x), and h(x).


Property 1: f(x) = -f(x)

This property tells us that the function f(x) is an odd function, meaning it is symmetric about the origin (0,0). This implies that for any x, f(x) = -f(x) = 0. Therefore, f(x) must be identically zero.

So, we can conclude that f(x) = 0 for all x.


Property 2: f(x) = g(x)

Since f(x) = 0, we can substitute this into the equation f(x) = g(x) to get g(x) = 0 for all x.

So, we can conclude that g(x) = 0 for all x.


Property 3: h(x) = [f(x)]^2 * [g(x)]^2

Using the values of f(x) and g(x) obtained from the previous properties, we can simplify the expression for h(x) as follows:

h(x) = [0]^2 * [0]^2

h(x) = 0


Calculating h(10)

Now that we know h(x) is identically zero, we can easily find the value of h(10) by substituting x = 10 into the expression for h(x):

h(10) = 0


Conclusion

The value of h(10) is zero. Therefore, the correct answer is option 'D' (None of these).

Solving by variation of parameter y" - 2y'+ y = exlog x, the value of wronskion W is
  • a)
    e2n
  • b)
    2
  • c)
    e-2n
  • d)
    None of these
Correct answer is option 'D'. Can you explain this answer?

Chirag Verma answered
A.E. for given differential equation is 
m2 - 2m + 1 = 0
implies 
Now, 
are two linearly independent solutions of given differential equation

Which of the following is not an exact differential?
  • a)
  • b)
  • c)
  • d)
    All of the above are exact differentials
Correct answer is option 'A'. Can you explain this answer?

Chirag Verma answered
The expression in (a) is not exact. Hence (a) is the correct answer.
Remark : Note that expression in (a)

Comments about Integrating Factor 
Definition : Suppose that tne differential equation. M dx + N dy = 0    ...(i)
is not exact but the differential equation

is exact, where μ =  F(x, y) for a suitably chosen function F. Then μ is called an integrating factor of the differential equation (i).
Ex. : The differential equation
(3ty + 4xy2) dx + (2x + 3x2y) dy = 0    ...(ii)
is not exact. But if we choose
μ = x2 y.
then the differential equation

becomes exact.
∴ μ = x2 y is an integrating factor of (ii).
Rules for finding an Integrating Factor :
Rule : 1: If Mx ± Ny ≠ 0. and is homogeneous in x and y then the integrating factor is

Ex. : Consider the differential equation
(x2y - 2xy2)dx- (x3 - 3x2y)dy = 0    ...(iii)
verify that
(i) equation (iii) is noi exact and (ii)

Intergrating factor is given by

Hence the equation

should be exact. Now equation (iv) can be written as 


Rule : 2. If 

is a function of x alone, say f(x), then the integrating factor μ is given by

Rule : 3. If

is a function of y alone, say φ(y), then the integrating factor μ is given by

For a partial differential equation, in a function φ (x, y) and two variables x, y, what is the form obtained after separation of variables is applied?
  • a)
    Φ (x, y) = X(x) + Y(y)
  • b)
    Φ (x, y) = X(x) - Y(y)
  • c)
    Φ (x, y) = X(x) / Y(y)
  • d)
    Φ (x, y) = X(x)Y(y)
Correct answer is option 'D'. Can you explain this answer?

Veda Institute answered
The method of separation of variables relies upon the assumption that a function of the form,
Φ (x, y) = X(x)Y(y)
will be a solution to a linear homogeneous partial differential equation in x and y. This is called a product solution and provided the boundary conditions are also linear and homogeneous this will also satisfy the boundary conditions.

Consider the differential equation ( x + y + 1) dx + (2x + 2y + 1) dy = 0. Which of the following statements is true?
  • a)
    The differential equation is linear
  • b)
    The differential equation is exact
  • c)
    ex + y is an integrating factor of the differential equation
  • d)
    A suitable substitution transforms the differentiable equation to the variable separable form
Correct answer is option 'D'. Can you explain this answer?

Dronacharya Institute answered
To determine which statement is true about the given differential equation, let's analyze each option:

a) The differential equation is linear:
A linear differential equation is one in which the dependent variable and its derivatives appear only to the first power and are not multiplied together. In the given equation, both x and y appear to the first power, but they are multiplied together in the terms (xy)dx and (2xy)dy. Therefore, the given differential equation is not linear.

b) The differential equation is exact:
A differential equation is exact if it can be written in the form M(x, y)dx + N(x, y)dy = 0, where M and N are functions of x and y, and ∂M/∂y = ∂N/∂x. Let's check if this condition holds for the given equation:
M(x, y) = x + y
N(x, y) = 2x + 2y
∂M/∂y = 1
∂N/∂x = 2
Since ∂M/∂y is not equal to ∂N/∂x, the given differential equation is not exact.

c) ex^y is an integrating factor of the differential equation:
An integrating factor is a function that can be multiplied to a differential equation to make it exact. In this case, the integrating factor would need to be able to transform the given equation into an exact one. However, as we determined in the previous option, the given equation is not exact. Therefore, ex^y is not an integrating factor.

d) A suitable substitution transforms the differential equation to the variable separable form:
Variable separable form means that the equation can be written in the form f(x)dx = g(y)dy, where f(x) and g(y) are functions of x and y, respectively. To determine if a suitable substitution can transform the given equation into this form, let's make the substitution x = u and y = v, where u and v are functions of a new variable t (u = u(t) and v = v(t)). Substituting these into the given equation, we have:
(uv)du + (2uv)dv = 0
Factoring out uv, we get:
uv(du + 2dv) = 0
Since this equation can be separated into f(u)du = -2g(v)dv, where f(u) = uv and g(v) = -v, we can conclude that a suitable substitution transforms the differential equation to the variable separable form.

Therefore, the correct answer is option d) A suitable substitution transforms the differential equation to the variable separable form.

If e-x, xe-x are solutions of y" + ay' + by = 0, then
  • a)
    a = 0, b = 1
  • b)
    a = 1, b = 2
  • c)
    a = 2, b = 1
  • d)
    a = -1, b = 0
Correct answer is option 'C'. Can you explain this answer?

Aanya Sharma answered
If $e^{-x}$ and $xe^{-x}$ are solutions of the differential equation $y'' - y' = 0$, then the general solution of the differential equation is given by $y(x) = C_1 e^{-x} + C_2 xe^{-x}$, where $C_1$ and $C_2$ are constants.

The equation xdy = ydx represents the family of
  • a)
    Circles
  • b)
    Ellipses
  • c)
    Hyperbolas
  • d)
    Straight lines
Correct answer is option 'D'. Can you explain this answer?

Veda Institute answered
The differential equation is x dy = y dx

This represents the family of straight lines passing through the origin.

Which of the following solutions of the differential equation  is a singular solution?
  • a)
  • b)
    3y = 9x + 3
  • c)
    y = 5 x + 1
  • d)
    y2 = 20x
Correct answer is option 'D'. Can you explain this answer?

Veda Institute answered
Some Definitions.
General solution : A solution of the differential equation of order n containing n essential arbitrary constants is called a general solution of the differential equation.
Particular solution : A solution which is obtained from the general solution by giving particular values to the arbitrary constants is called a Particular solution.
Singular solution : A solution which can not be obtained from the general solution of the differential equation by any chioice of the n essential arbitrary constants is ealled a Singular solution.
Ex : Consider the differential eauation.
 ..(i)
We have
General solution :    y = (x + c)2
Particular solutions : y = x : For c = 0
y = x + 10 : For c = 10 etc.
Singular solutions : y = 0 Note that y = 0 satisfies the differential equation (i) and can not be obtained from the general solution by giving any value to c.
The given differential equation is


Its general solution is given by

But the solution y2 - 20x (verify that it satisfies equation (ii)) but it can not be obtained from the general solution given by (iii) for any value of c. Therefore, y2 -= 20x is a singular solution of differential equation (ii).

The degree of a differential equation is defined as the
  • a)
    Highest of the orders of the differential coefficients occuring in it
  • b)
    Highest power of the highest order differential coefficient occurring in it
  • c)
    Any power of the highest order differential coefficient occurring in it
  • d)
    Highest power among the powers of the dif ferential coefficients occurring in it
Correct answer is option 'B'. Can you explain this answer?

Veda Institute answered
Definition : The degree of a differential equation is defined as the highest power of the highest order derivative involved in the differential, where the equation has been made rational and integral as far as the derivatives are concerned. 
degree of DE (1) = 2
degree of DE (2) = 1
degree of DE (3) = 1
An Important Example : Find the degree of the differential equation.

In order to rationalize this equation, we first square this equation and get

Now clearly the degree of the differential equation is 2.
Thus statement (b) gives the correct definition of the degee of a differential equation.

Solution of the differential equation xy' + sin 2y = x3 sin2 y is 
  • a)
    cot y = -x3 + cx2
  • b)
    2cot y = -x3 + 2cx2
  • c)
    tan y = -x3 + cx2
  • d)
    2tan y = x3 + 2cx2
Correct answer is option 'A'. Can you explain this answer?

Qamar Siddiqui answered
Given Differential Equation:
The given differential equation is xy + sin(2y) = x^3sin(2y).

Solution:
To solve this differential equation, we can use the method of separating variables.

Step 1: Separating Variables
- Move all terms involving y to one side and terms involving x to the other side.
- Divide by sin(2y) to isolate terms involving y and divide by x^3 to isolate terms involving x.

Step 2: Integrating both sides
- Integrate both sides with respect to the respective variables.

Step 3: Solve for the Constant of Integration
- Use the initial conditions or boundary conditions if provided to find the constant of integration.

Step 4: Final Solution
- After finding the constant of integration, rewrite the solution with the constant included.
Therefore, the solution to the given differential equation is cot y = -x^3 + cx^2, where c is the constant of integration.

A differential equation of first degree
  • a)
    Is always linear
  • b)
    Is of first order
  • c)
    May or may not be linear
  • d)
    Is never of first order but is linear always
Correct answer is option 'C'. Can you explain this answer?

Chirag Verma answered
(i)(a) is not correct because a differential equation of first degree may not be linear. For example, the differential equation.

is of first degree but non-linear, 
(ii) (b) is not correct because a differential equation of first, degree is not necessarily of first order. The differential equation (i) is of first, degree but of second order.
(iii) (d) is not correct because a differential equation of first degree may be of first order and may not be linear.

If the general solutions of a differential equation are (y + c)2 = cx, where c is an arbitrary constant, then the order and degree of differential equation is  
  • a)
    1, 2
  • b)
    2, 1
  • c)
    1, 3 
  • d)
     None of these
Correct answer is option 'A'. Can you explain this answer?

Chirag Verma answered
There will only one constant in the first-order differential equation. Differentiating the given equation.

 
Putting the value of c in Eq. (1) and simplifying we will get a first-order and second-degree equation. Hence, (A) is the correct answer.

The orthogonal trajectories of the given family of curves y = cx2 are given by
  • a)
    x2 + cy2 = constant
  • b)
    x2 + ky2 = constant
  • c)
    kx2 + y2 = constant 
  • d)
    x2 - ky2 = constant
Correct answer is option 'B'. Can you explain this answer?

Chirag Verma answered
Then given family of curves is
y = cxk  ....(i)
.v —
Let us first find the differential equation satisfied by the family (i). For this we differentiate (i) w.r. t. 

∴ The differential equation of the orthogonal trajectories will be obtained on replacing 
⇒ Orthogonal trajectories are given by

Consider the differential equation 2 cos (y2)dx - xy sin (y2)dy = 0
  • a)
    ex is an integrating factor
  • b)
    e-x is an integrating factor
  • c)
    3x is an integrating factor 
  • d)
    x3 is an integrating factor
Correct answer is option 'D'. Can you explain this answer?

Kanika Verma answered
Given: The differential equation 2 cos (y^2)dx - xy sin (y^2)dy = 0

To solve this differential equation, we can use the method of integrating factors. An integrating factor is a function that we can multiply to the entire equation to make it exact.

Step 1: Check if the equation is exact:
An equation of the form M(x, y)dx + N(x, y)dy = 0 is exact if and only if ∂M/∂y = ∂N/∂x.

Let's find the partial derivatives:
∂M/∂y = -4ycos(y^2)
∂N/∂x = -ysin(y^2)

Since ∂M/∂y is not equal to ∂N/∂x, the given differential equation is not exact.

Step 2: Find the integrating factor:
To find the integrating factor, we can use the formula: integrating factor = e^(∫(∂N/∂x - ∂M/∂y)/N dx).

In this case, ∂M/∂y - ∂N/∂x = -4ycos(y^2) - (-ysin(y^2)) = -4ycos(y^2) + ysin(y^2).

We need to find the integral of (-4ycos(y^2) + ysin(y^2))/(-xy sin(y^2)) dx.

Simplifying the expression, we get (∫(4cos(y^2)/y - sin(y^2))/x dx.

The integral (∫4cos(y^2)/y dx) can be solved using the substitution method, and the integral (∫sin(y^2)/x dx) can be solved using the logarithmic method.

After finding the integral, we get the integrating factor as x^4/y^4.

Step 3: Multiply the integrating factor to the equation:
Multiplying the integrating factor x^4/y^4 to the given differential equation, we get:

(2x^3/y^4)cos(y^2)dx - (x^5/y^3)sin(y^2)dy = 0

This equation is exact, and we can solve it by finding the potential function or by using other methods.

Therefore, the correct answer is option D) x^3 is an integrating factor.

Which one of the following is a general solution of the differential equation y = 2px + p2y ?
  • a)
    2yc - x2 + c2 = 0
  • b)
    2xc + y2 - c2 = 0
  • c)
    2xc - y2 + c2 = 0
  • d)
    yc + x2 + c2 = 0
Correct answer is option 'C'. Can you explain this answer?

Arnav Chawla answered
To solve the given differential equation, we need to find a general solution that satisfies the equation.

The given differential equation is: y = 2px - p^2y

To solve this equation, we can rearrange it to isolate the terms involving y on one side:

y + p^2y = 2px

Factor out y on the left side:

y(1 + p^2) = 2px

Divide both sides by (1 + p^2):

y = (2px) / (1 + p^2)

Now, let's simplify this expression further.

We can rewrite 2px as 2xc since p represents the derivative of x with respect to y.

Therefore, the general solution of the given differential equation is:

y = (2xc) / (1 + p^2)

Now, let's rewrite this expression in terms of x and y.

p = dx/dy, so we can rewrite it as:

p = (dx/dy) = (dx)/(dy/dx) = (dx/dy) / (1/(dx/dy))

Therefore, p^2 = ((dx/dy) / (1/(dx/dy)))^2 = ((dx/dy)^2) / (1/(dx/dy))^2 = (dx^2)/(dy^2)

Substituting this back into the equation, we get:

y = (2xc) / (1 + (dx^2)/(dy^2))

Now, let's simplify this expression further.

Multiply the numerator and denominator by (dy^2):

y = (2xc * dy^2) / (dy^2 + dx^2)

Now, we can rewrite dy^2 + dx^2 as d(x^2 + y^2):

y = (2xc * dy^2) / d(x^2 + y^2)

Cancel out the d terms:

y = (2xc * y^2) / (x^2 + y^2)

Multiply both sides by (x^2 + y^2):

y(x^2 + y^2) = 2xc * y^2

Rearrange the terms:

y(x^2 + y^2 - 2xc * y) = 0

This is the general solution of the given differential equation.

Therefore, the correct option is C) 2xc - y^2 + c^2 = 0.

Orthogonal trajectories of the family of curves (x - 1)2 + y2 + 2ax = 0 are the solution of the differential equation
  • a)
  • b)
  • c)
    x2 - y2 - 1 - 2xydy/dx = 0(d)
  • d)
Correct answer is option 'A'. Can you explain this answer?

Chirag Verma answered
we have (x - 1)2 + y2 + 2ax = 0 ...(i)
Differentiating w.r.t. x, we get 
2(x - 1) + 2y y' + 2a = 0
implies 

Putting expression of a in eq. (i), we get
(x - 1)2 + y2 - 2x((x - 1) + yy') = 0
implies x2 - 2x + 1 + y2 - 2x2 + 2x + 2xyy' = 0 

Orthogonal trajectories are given by

Consider the following statements
I. A singular solution of differential equation satisfies the differential equation but is not a particular solution of the equation.
II. If T(x, y) = 0 is the equation of the tac-locus, then
T(x, y ) is a factor of the P-discriminant.
III. T(x, y ) is a factor of a c-discriminant.
IV. Cusp-locus has two distinct tangent.
Choose the correct answer.
  • a)
    Only I is true
  • b)
    I, II and IV are true
  • c)
    All are true
  • d)
    No one is true
Correct answer is option 'B'. Can you explain this answer?

Saanvi Das answered
I. A singular solution of a differential equation satisfies the differential equation but is not a particular solution of the equation.
A singular solution of a differential equation is a solution that cannot be obtained from the general solution by specifying values for any arbitrary constants. It satisfies the differential equation, but it is not a particular solution because it does not correspond to any specific initial conditions or boundary conditions. Therefore, statement I is true.

II. If T(x, y) = 0 is the equation of the tac-locus, then T(x, y) is a factor of the P-discriminant.
The tac-locus is the locus of points where the tangent to a given curve is parallel to a given line. If T(x, y) = 0 is the equation of the tac-locus, it means that the curve satisfies the condition of having a parallel tangent to the given line. The P-discriminant is a discriminant that determines the nature of the points on a curve. If T(x, y) = 0 is the equation of the tac-locus, then T(x, y) must be a factor of the P-discriminant. Therefore, statement II is true.

III. T(x, y) is a factor of a c-discriminant.
The c-discriminant is a discriminant that determines the nature of the critical points on a curve. There is no direct relationship between the tac-locus equation T(x, y) = 0 and the c-discriminant. Therefore, statement III is false.

IV. Cusp-locus has two distinct tangents.
A cusp is a point on a curve where the curve has a sharp corner or a point of self-intersection. At the cusp point, the curve has two distinct tangents that are not parallel to each other. Therefore, statement IV is true.

Based on the explanations above, we can conclude that statements I, II, and IV are true. Therefore, the correct answer is option 'B'.

Which of the following differential equations is not of degree 1?
  • a)
  • b)
  • c)
  • d)
Correct answer is option 'B'. Can you explain this answer?

Veda Institute answered
i. Clearly degree of DE in (a) = 1
ii. DE in (b) must be first squared to make it. rational. Then differential equation becomes.

or

so that the highest power of the highest order derivative is 2. Hence its degree is 2.
iii. The differential equations in (c) and (d) must be squared first to make them rational.
Then we find that their degree is 1.
∴ The differential equations in (a), (c) and (d) are each of degree 1 and in (b) is of degree 2.

General solution of pde given below is (y2+ z2+ x2 )p - 2xyq + 2xz = 0
  • a)
  • b)
  • c)
  • d)
Correct answer is option 'A'. Can you explain this answer?

Chirag Verma answered
ANSWER :- a
Solution :- Lagrange auxiliary equations are :-
dx/(y^2 + z^2 - x^2) = -dy/(-2xy) = -dz/(2xz)
Taking the last two members, we get
dy/y = dz/z
Integrating log y = logz + logc1
=y/z = c1
Using ,multipliers x,y,z we get (xdx + ydy + zdz)/-x(x^2 + y^2 + z^2) 
(xdx + ydy + zdz)/-x(x^2 + y^2 + z^2) = dx/(-2xz)
2(xdx + ydy + zdz)/(x^2 + y^2 + z^2) = dz/z
Integrating, log(x^2 + y^2 + z^2) = logz + logc2
(x^2 + y^2 + z^2) = zc2
Hence the required solution is f(c1,c2) = 0
= f(y/z, (x^2 + y^2 + z^2)/z) = 0

The general solution of the differential equation (y - px) (p  - 1) = p is given by
  • a)
  • b)
    y2 = 4cx
  • c)
  • d)
Correct answer is option 'D'. Can you explain this answer?

Veda Institute answered
Proof: The given differential equation is
(y - px) (p - i) = p ...... (i)
  ......(ii)
Differentiating (ii) w.r. t. x, we get



⇒ p = c     .......(iii)
Eliminating p from equations (ii) and (iii),

Which one of the following is an ordinary differential equation?
  • a)
  • b)
  • c)
  • d)
Correct answer is option 'D'. Can you explain this answer?

Chirag Verma answered
Definition : The equations involving derivatives of one or more dependent variables with respect to a single independent variable are called ordinary differential equations.
Ex.:

are the examples of ordinary differential equations where y is dependent variable and x is independent variable.

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