>
Exams
>
Mathematics
>
Differential equations
>
the solution of the differential equation dydx sec
Question:
The solution of the differential equation
d
y
d
x
=
sec
(
y
x
)
+
y
x
is:
MHT CET
Updated On:
Jul 27, 2024
(A)
cos
(
y
x
)
=
log
(
c
x
)
(B)
sin
(
x
y
)
=
log
(
c
x
)
(C)
sin
(
y
x
)
=
log
(
c
x
)
(D) None of these
Hide Solution
Verified By Collegedunia
The Correct Option is
C
Solution and Explanation
Explanation:
Given,
d
y
d
x
=
sec
(
y
x
)
+
y
x
…
…
(
1
)
Let,
y
x
=
t
⇒
y
=
x
t
Differentiating with respect to
x
, we get
d
y
d
x
=
x
×
d
t
d
x
+
t
×
d
d
x
x
We known that:
d
d
x
x
y
=
y
×
d
d
x
x
+
x
×
d
d
x
y
So,
d
y
d
x
=
x
×
d
t
d
x
+
t
Now, Putting these value in equation
(
1
)
, we get
x
d
t
d
x
+
t
=
sec
t
+
t
⇒
x
d
t
d
x
=
sec
t
⇒
d
t
sec
t
=
d
x
x
Integrating both sides, we get
∫
d
t
sec
t
=
∫
d
x
x
⇒
∫
cos
t
d
t
=
∫
d
x
x
⇒
sin
t
=
log
x
+
log
c
⇒
sin
t
=
log
(
c
x
)
(
∵
log
m
+
log
n
=
log
(
m
n
)
)
∴
sin
(
y
x
)
=
log
(
c
x
)
Hence, the correct option is (C).
Download Solution in PDF
Was this answer helpful?
0
0
Top Questions on Differential equations
Let \( y = f(x) \) be the solution of the differential equation
\[\frac{dy}{dx} + \frac{xy}{x^2 - 1} = \frac{x^6 + 4x}{\sqrt{1 - x^2}}, \quad -1 < x < 1\]
such that \( f(0) = 0 \). If
\[6 \int_{-1/2}^{1/2} f(x)dx = 2\pi - \alpha\]
then \( \alpha^2 \) is equal to __________.
JEE Main - 2025
Mathematics
Differential equations
View Solution
Let \( y = y(x) \) be the solution of the differential equation \( \frac{dy}{dx} + 3(\tan^2 x) y + 3y = \sec^2 x \), with \( y(0) = \frac{1}{3} + e^3 \). Then \( y\left(\frac{\pi}{4}\right) \) is equal to
JEE Main - 2025
Mathematics
Differential equations
View Solution
Let g be a differentiable function such that $ \int_0^x g(t) dt = x - \int_0^x tg(t) dt $, $ x \ge 0 $ and let $ y = y(x) $ satisfy the differential equation $ \frac{dy}{dx} - y \tan x = 2(x+1) \sec x g(x) $, $ x \in \left[ 0, \frac{\pi}{2} \right) $. If $ y(0) = 0 $, then $ y\left( \frac{\pi}{3} \right) $ is equal to
JEE Main - 2025
Mathematics
Differential equations
View Solution
If a curve $ y = y(x) $ passes through the point $ \left(1, \frac{\pi}{2}\right) $ and satisfies the differential equation
$$ (7x^4 \cot y - e^x \csc y) \frac{dx}{dy} = x^5, \quad x \geq 1, \text{ then at } x = 2, \text{ the value of } \cos y \text{ is:} $$
JEE Main - 2025
Mathematics
Differential equations
View Solution
General solution of the differential equation
\[ \frac{dy}{dx} + y \tan x = \sec x \quad \text{is:} \]
KCET - 2025
Mathematics
Differential equations
View Solution
View More Questions
Questions Asked in MHT CET exam
A 2 kg object is in a gravitational field where the acceleration due to gravity is \( 9.8 \, \text{m/s}^2 \). What is the gravitational potential energy of the object at a height of \( 5 \, \text{m} \)?
MHT CET - 2025
work, energy and power
View Solution
A sound wave has a frequency of $ 440 \, \text{Hz} $. What is its time period?
MHT CET - 2025
Waves
View Solution
If \( \mathbf{a} = 2\hat{i} + 2\hat{j} + 3\hat{k}, \mathbf{b} = -\hat{i} + 2\hat{j} + \hat{k} \) and \( \mathbf{c} = 3\hat{i} + \hat{j} \) are the vectors such that \( \mathbf{a} + \lambda \mathbf{b} \) is perpendicular to \( \mathbf{c} \), then the value of \( \lambda \) is:
MHT CET - 2025
Vectors
View Solution
What is the empirical formula of a compound containing 40% sulfur and 60% oxygen by mass?
MHT CET - 2025
Chemical bonding and molecular structure
View Solution
A car accelerates uniformly from rest to a velocity of \( 25 \, \text{m/s} \) in \( 10 \, \text{seconds} \). What is the acceleration of the car?
MHT CET - 2025
Motion in a straight line
View Solution
View More Questions