Advertisements
Advertisements
प्रश्न
Advertisements
उत्तर
We have,
\[ \cos^2 \left( x - 2y \right) = 1 - 2\frac{dy}{dx}\]
\[ \Rightarrow 2\frac{dy}{dx} = 1 - \cos^2 \left( x - 2y \right)\]
\[\text{Let }x - 2y = v\]
\[ \Rightarrow 1 - 2\frac{dy}{dx} = \frac{dv}{dx}\]
\[ \Rightarrow 2\frac{dy}{dx} = 1 - \frac{dv}{dx}\]
\[ \therefore 1 - \frac{dv}{dx} = 1 - \cos^2 v\]
\[ \Rightarrow \frac{dv}{dx} = \cos^2 v\]
\[ \Rightarrow \sec^2 v dv = dx\]
Integrating both sides, we get
\[\int \sec^2 v dv = \int dx\]
\[ \Rightarrow \tan v = x - C\]
\[ \Rightarrow \tan\left( x - 2y \right) = x - C\]
\[ \Rightarrow x = \tan\left( x - 2y \right) + C\]
APPEARS IN
संबंधित प्रश्न
Solve the equation for x: `sin^(-1) 5/x + sin^(-1) 12/x = π/2, x ≠ 0`
Find the differential equation of all the parabolas with latus rectum '4a' and whose axes are parallel to x-axis.
Verify that y2 = 4a (x + a) is a solution of the differential equations
\[y\left\{ 1 - \left( \frac{dy}{dx} \right)^2 \right\} = 2x\frac{dy}{dx}\]
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[x\frac{dy}{dx} + y = y^2\]
|
\[y = \frac{a}{x + a}\]
|
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 |
|
\[y = \left( \frac{dy}{dx} \right)^2\]
|
\[y = \frac{1}{4} \left( x \pm a \right)^2\]
|
xy (y + 1) dy = (x2 + 1) dx
(ey + 1) cos x dx + ey sin x dy = 0
(y2 + 1) dx − (x2 + 1) dy = 0
dy + (x + 1) (y + 1) dx = 0
Solve the following differential equation:
\[y e^\frac{x}{y} dx = \left( x e^\frac{x}{y} + y^2 \right)dy, y \neq 0\]
(x + y) (dx − dy) = dx + dy
Solve the following initial value problem:-
\[\frac{dy}{dx} + 2y = e^{- 2x} \sin x, y\left( 0 \right) = 0\]
Solve the following initial value problem:-
\[x\frac{dy}{dx} - y = \left( x + 1 \right) e^{- x} , y\left( 1 \right) = 0\]
If the marginal cost of manufacturing a certain item is given by C' (x) = \[\frac{dC}{dx}\] = 2 + 0.15 x. Find the total cost function C (x), given that C (0) = 100.
Find the equation of the curve which passes through the point (2, 2) and satisfies the differential equation
\[y - x\frac{dy}{dx} = y^2 + \frac{dy}{dx}\]
Find the equation of the curve passing through the point (0, 1) if the slope of the tangent to the curve at each of its point is equal to the sum of the abscissa and the product of the abscissa and the ordinate of the point.
The solution of the differential equation y1 y3 = y22 is
Which of the following is the integrating factor of (x log x) \[\frac{dy}{dx} + y\] = 2 log x?
Determine the order and degree of the following differential equations.
| Solution | D.E |
| y = aex + be−x | `(d^2y)/dx^2= 1` |
Solve the following differential equation.
`dy /dx +(x-2 y)/ (2x- y)= 0`
Solve the following differential equation.
`(x + a) dy/dx = – y + a`
Solve the differential equation:
`e^(dy/dx) = x`
`dy/dx = log x`
Select and write the correct alternative from the given option for the question
Differential equation of the function c + 4yx = 0 is
Solve the differential equation xdx + 2ydy = 0
For the differential equation, find the particular solution
`("d"y)/("d"x)` = (4x +y + 1), when y = 1, x = 0
Choose the correct alternative:
Solution of the equation `x("d"y)/("d"x)` = y log y is
Choose the correct alternative:
General solution of `y - x ("d"y)/("d"x)` = 0 is
State whether the following statement is True or False:
The integrating factor of the differential equation `("d"y)/("d"x) - y` = x is e–x
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?
What distinguishes an ordinary differential equation from other types of differential equations?
