Advertisements
Advertisements
प्रश्न
Advertisements
उत्तर
We have,
\[\left( x - 1 \right)\frac{dy}{dx} = 2 xy\]
\[ \Rightarrow \left( x - 1 \right)dy = 2xy dx\]
\[ \Rightarrow \frac{2x}{\left( x - 1 \right)}dx = \frac{1}{y}dy\]
Integrating both sides, we get
\[2\int\frac{x}{\left( x - 1 \right)}dx = \int\frac{1}{y}dy\]
\[ \Rightarrow 2\int\frac{x - 1 + 1}{x - 1}dx = \int\frac{1}{y}dy\]
\[ \Rightarrow 2\int dx + 2\int\frac{1}{x - 1}dx = \int\frac{1}{y}dy\]
\[ \Rightarrow 2x + 2 \log\left| x - 1 \right| = \log\left| y \right| + C\]
APPEARS IN
संबंधित प्रश्न
Hence, the given function is the solution to the given differential equation. \[\frac{c - x}{1 + cx}\] is a solution of the differential equation \[(1+x^2)\frac{dy}{dx}+(1+y^2)=0\].
xy (y + 1) dy = (x2 + 1) dx
y (1 + ex) dy = (y + 1) ex dx
Find the solution of the differential equation cos y dy + cos x sin y dx = 0 given that y = \[\frac{\pi}{2}\], when x = \[\frac{\pi}{2}\]
The volume of a spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 units. Find the radius of the balloon after `t` seconds.
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).
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:-
\[dy = \cos x\left( 2 - y\text{ cosec }x \right)dx\]
Find the equation of the curve which passes through the origin and has the slope x + 3y− 1 at any point (x, y) on it.
Find the equation of the curve which passes through the point (1, 2) and the distance between the foot of the ordinate of the point of contact and the point of intersection of the tangent with x-axis is twice the abscissa of the point of contact.
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.
The rate of increase of bacteria in a culture is proportional to the number of bacteria present and it is found that the number doubles in 6 hours. Prove that the bacteria becomes 8 times at the end of 18 hours.
Find the solution of the differential equation
\[x\sqrt{1 + y^2}dx + y\sqrt{1 + x^2}dy = 0\]
The solution of the differential equation \[\frac{dy}{dx} = \frac{ax + g}{by + f}\] represents a circle when
The differential equation of the ellipse \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = C\] is
The integrating factor of the differential equation \[\left( 1 - y^2 \right)\frac{dx}{dy} + yx = ay\left( - 1 < y < 1 \right)\] is ______.
In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-
y = ex + 1 y'' − y' = 0
Choose the correct option from the given alternatives:
The solution of `1/"x" * "dy"/"dx" = tan^-1 "x"` is
Determine the order and degree of the following differential equations.
| Solution | D.E. |
| y = 1 − logx | `x^2(d^2y)/dx^2 = 1` |
For the following differential equation find the particular solution.
`dy/ dx = (4x + y + 1),
when y = 1, x = 0
Solve the following differential equation.
xdx + 2y dx = 0
Solve the following differential equation.
`(x + y) dy/dx = 1`
Choose the correct alternative.
The differential equation of y = `k_1 + k_2/x` is
The differential equation of `y = k_1e^x+ k_2 e^-x` is ______.
y dx – x dy + log x dx = 0
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 the differential equation sec2y tan x dy + sec2x tan y dx = 0
Solve the following differential equation
`yx ("d"y)/("d"x)` = x2 + 2y2
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
An appropriate substitution to solve the differential equation `"dx"/"dy" = (x^2 log(x/y) - x^2)/(xy log(x/y))` is ______.
