Advertisements
Advertisements
प्रश्न
dy + (x + 1) (y + 1) dx = 0
Advertisements
उत्तर
We have,
\[dy + \left( x + 1 \right)\left( y + 1 \right) dx = 0\]
\[ \Rightarrow dy = - \left( x + 1 \right)\left( y + 1 \right) dx\]
\[ \Rightarrow \frac{1}{y + 1}dy = - \left( x + 1 \right) dx\]
Integrating both sides, we get
\[\int\frac{1}{y + 1}dy = - \int\left( x + 1 \right) dx\]
\[ \Rightarrow \log \left| y + 1 \right| = - \frac{x^2}{2} - x + C\]
\[ \Rightarrow \log \left| y + 1 \right| + \frac{x^2}{2} + x = C\]
\[\text{ Hence, }\log \left| y + 1 \right| + \frac{x^2}{2} + x =\text{ C is the required solution . }\]
APPEARS IN
संबंधित प्रश्न
Solve the equation for x: `sin^(-1) 5/x + sin^(-1) 12/x = π/2, x ≠ 0`
Form the differential equation representing the family of ellipses having centre at the origin and foci on x-axis.
Show that Ax2 + By2 = 1 is a solution of the differential equation x \[\left\{ y\frac{d^2 y}{d x^2} + \left( \frac{dy}{dx} \right)^2 \right\} = y\frac{dy}{dx}\]
Solve the following differential equation:
(xy2 + 2x) dx + (x2 y + 2y) dy = 0
Solve the differential equation \[\frac{dy}{dx} = \frac{2x\left( \log x + 1 \right)}{\sin y + y \cos y}\], given that y = 0, when x = 1.
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.
(y2 − 2xy) dx = (x2 − 2xy) dy
3x2 dy = (3xy + y2) dx
Experiments show that radium disintegrates at a rate proportional to the amount of radium present at the moment. Its half-life is 1590 years. What percentage will disappear in one year?
Show that the equation of the curve whose slope at any point is equal to y + 2x and which passes through the origin is y + 2 (x + 1) = 2e2x.
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.
Write the differential equation obtained by eliminating the arbitrary constant C in the equation x2 − y2 = C2.
Which of the following transformations reduce the differential equation \[\frac{dz}{dx} + \frac{z}{x}\log z = \frac{z}{x^2} \left( \log z \right)^2\] into the form \[\frac{du}{dx} + P\left( x \right) u = Q\left( x \right)\]
The differential equation
\[\frac{dy}{dx} + Py = Q y^n , n > 2\] can be reduced to linear form by substituting
Which of the following differential equations has y = C1 ex + C2 e−x as the general solution?
Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.
Solve the following differential equation.
`xy dy/dx = x^2 + 2y^2`
Solve the following differential equation.
`x^2 dy/dx = x^2 +xy - y^2`
Solve the differential equation:
dr = a r dθ − θ dr
`dy/dx = log x`
Solve the following differential equation `("d"y)/("d"x)` = x2y + y
The solution of differential equation `x^2 ("d"^2y)/("d"x^2)` = 1 is ______
Verify y = log x + c is the solution of differential equation `x ("d"^2y)/("d"x^2) + ("d"y)/("d"x)` = 0
The integrating factor of the differential equation `"dy"/"dx" (x log x) + y` = 2logx is ______.
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?
Solve the differential equation
`y (dy)/(dx) + x` = 0
Solve the differential equation `dy/dx + xy = xy^2` and find the particular solution when y = 4, x = 1.
