Advertisements
Advertisements
प्रश्न
(ey + 1) cos x dx + ey sin x dy = 0
Advertisements
उत्तर
We have,
\[\left( e^y + 1 \right) \cos x dx + e^y \sin x dy = 0\]
\[ \Rightarrow e^y \sin x dy = - \left( e^y + 1 \right) \cos x dx\]
\[ \Rightarrow \frac{e^y}{e^y + 1}dy = - \frac{\cos x}{\sin x}dx\]
\[ \Rightarrow \frac{e^y}{e^y + 1}dy = - \cot x dx\]
Integrating both sides, we get
\[\int\frac{e^y}{e^y + 1}dy = - \int\cot x dx\]
\[\text{ Putting }e^y + 1 = t,\text{ we get }\]
\[ e^y dy = dt\]
\[ \therefore \int\frac{dt}{t} = - \int\cot x dx\]
\[ \Rightarrow \log\left| t \right| = - \log \left| \sin x \right| + \log C \]
\[ \Rightarrow \log \left| e^y + 1 \right| + \log \left| \sin x \right| = \log C\]
\[ \Rightarrow \log\left| \left( e^y + 1 \right) \sin x \right| = \log C\]
\[ \Rightarrow \left( e^y + 1 \right) \sin x = C\]
\[ \Rightarrow \left( e^y + 1 \right) \sin x = C\]
\[\text{ Hence, }\left( e^y + 1 \right) \sin x = C\text{ is the required solution}. \]
APPEARS IN
संबंधित प्रश्न
Differential equation \[\frac{d^2 y}{d x^2} + y = 0, y \left( 0 \right) = 0, y' \left( 0 \right) = 1\] Function y = sin x
(1 + x) (1 + y2) dx + (1 + y) (1 + x2) dy = 0
dy + (x + 1) (y + 1) dx = 0
In a bank principal increases at the rate of r% per year. Find the value of r if ₹100 double itself in 10 years (loge 2 = 0.6931).
In a culture the bacteria count is 100000. The number is increased by 10% in 2 hours. In how many hours will the count reach 200000, if the rate of growth of bacteria is proportional to the number present.
\[x^2 \frac{dy}{dx} = x^2 + xy + y^2 \]
3x2 dy = (3xy + y2) dx
Find the particular solution of the differential equation \[\frac{dy}{dx} = \frac{xy}{x^2 + y^2}\] given that y = 1 when x = 0.
Define a differential equation.
Write the differential equation obtained by eliminating the arbitrary constant C in the equation x2 − y2 = C2.
The equation of the curve whose slope is given by \[\frac{dy}{dx} = \frac{2y}{x}; x > 0, y > 0\] and which passes through the point (1, 1) is
The differential equation of the ellipse \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = C\] is
The differential equation \[x\frac{dy}{dx} - y = x^2\], has the general solution
Which of the following differential equations has y = C1 ex + C2 e−x as the general solution?
y2 dx + (x2 − xy + y2) dy = 0
Verify that the function y = e−3x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + \frac{dy}{dx} - 6y = 0.\]
In each of the following examples, verify that the given function is a solution of the corresponding differential equation.
| Solution | D.E. |
| y = ex | `dy/ dx= y` |
Solve the following differential equation.
`y^3 - dy/dx = x dy/dx`
For each of the following differential equations find the particular solution.
`y (1 + logx)dx/dy - x log x = 0`,
when x=e, y = e2.
Select and write the correct alternative from the given option for the question
Bacterial increases at the rate proportional to the number present. If original number M doubles in 3 hours, then number of bacteria will be 4M in
Solve `("d"y)/("d"x) = (x + y + 1)/(x + y - 1)` when x = `2/3`, y = `1/3`
Solve the differential equation xdx + 2ydy = 0
Solve the following differential equation
`yx ("d"y)/("d"x)` = x2 + 2y2
The function y = cx is the solution of differential equation `("d"y)/("d"x) = y/x`
The integrating factor of the differential equation `"dy"/"dx" (x log x) + y` = 2logx is ______.
Solve: `("d"y)/("d"x) = cos(x + y) + sin(x + y)`. [Hint: Substitute x + y = z]
Solution of `x("d"y)/("d"x) = y + x tan y/x` is `sin(y/x)` = cx
There are n students in a school. If r % among the students are 12 years or younger, which of the following expressions represents the number of students who are older than 12?
