Advertisements
Advertisements
प्रश्न
Advertisements
उत्तर
We have,
\[ \sin^4 x\frac{dy}{dx} = \cos x\]
\[ \Rightarrow dy = \frac{\cos x}{\sin^4 x}dx\]
Integrating both sides, we get
\[ \Rightarrow \int dy = \int\frac{\cos x}{\sin^4 x}dx\]
\[ \Rightarrow y = \int\frac{\cos x}{\sin^4 x}dx\]
\[\text{ Putting }\sin x = t\]
\[ \Rightarrow \cos x dx = dt\]
\[ \therefore y = \int\frac{1}{t^4}dt\]
\[ = \frac{t^{- 3}}{- 3} + C\]
\[ = \frac{- \sin^{- 3} x}{3} + C\]
\[ = - \frac{1}{3} {cosec}^3 x + C \]
\[\text{ Hence, }y = - \frac{1}{3} {cosec}^3 x +\text{C is the solution to the given differential equation.}\]
APPEARS IN
संबंधित प्रश्न
Show that y = AeBx is a solution of the differential equation
Show that y = ax3 + bx2 + c is a solution of the differential equation \[\frac{d^3 y}{d x^3} = 6a\].
Verify that y = − x − 1 is a solution of the differential equation (y − x) dy − (y2 − x2) dx = 0.
Verify that y = log \[\left( x + \sqrt{x^2 + a^2} \right)^2\] satisfies the differential equation \[\left( a^2 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = 0\]
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[x^3 \frac{d^2 y}{d x^2} = 1\]
|
\[y = ax + b + \frac{1}{2x}\]
|
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[y = \left( \frac{dy}{dx} \right)^2\]
|
\[y = \frac{1}{4} \left( x \pm a \right)^2\]
|
Differential equation \[\frac{d^2 y}{d x^2} + y = 0, y \left( 0 \right) = 1, y' \left( 0 \right) = 1\] Function y = sin x + cos x
tan y dx + sec2 y tan x dy = 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 + y) (dx − dy) = dx + dy
Solve the following initial value problem:-
\[x\frac{dy}{dx} - y = \log x, y\left( 1 \right) = 0\]
Solve the following initial value problem:
\[x\frac{dy}{dx} + y = x \cos x + \sin x, y\left( \frac{\pi}{2} \right) = 1\]
The surface area of a balloon being inflated, changes at a rate proportional to time t. If initially its radius is 1 unit and after 3 seconds it is 2 units, find the radius after time t.
The rate of growth of a population is proportional to the number present. If the population of a city doubled in the past 25 years, and the present population is 100000, when will the city have a population of 500000?
The normal to a given curve at each point (x, y) on the curve passes through the point (3, 0). If the curve contains the point (3, 4), find its equation.
If sin x is an integrating factor of the differential equation \[\frac{dy}{dx} + Py = Q\], then write the value of P.
The solution of the differential equation y1 y3 = y22 is
The differential equation satisfied by ax2 + by2 = 1 is
The differential equation
\[\frac{dy}{dx} + Py = Q y^n , n > 2\] can be reduced to linear form by substituting
The integrating factor of the differential equation \[x\frac{dy}{dx} - y = 2 x^2\]
Solve the following differential equation.
(x2 − y2 ) dx + 2xy dy = 0
Solve the following differential equation.
`x^2 dy/dx = x^2 +xy - y^2`
Solve the following differential equation.
`dy/dx + 2xy = x`
Choose the correct alternative.
Bacteria increases at the rate proportional to the number present. If the original number M doubles in 3 hours, then the number of bacteria will be 4M in
Solve the differential equation xdx + 2ydy = 0
Choose the correct alternative:
General solution of `y - x ("d"y)/("d"x)` = 0 is
The differential equation of all non horizontal lines in a plane is `("d"^2x)/("d"y^2)` = 0
The differential equation (1 + y2)x dx – (1 + x2)y dy = 0 represents a family of:
