Advertisements
Advertisements
प्रश्न
Solve the following differential equation.
`dy/dx + 2xy = x`
Advertisements
उत्तर
`dy/dx + 2xy = x`
The given equation is of the form
`dy/dx + py = Q`
where, P = 2x and Q = x
∴ `I.F. = e^(intPdx) = e^ (int ^(2x dx) = e^(x^2)`
∴ Solution of the given equation is
y(I.F.) = `int Q ( I.F.) dx +c`
∴ `y e ^(x^2) int xe^(x^2) dx + c `
In R. H. S., put x2 = t
Differentiating w.r.t. x, we get
2x dx = dt
∴ `ye^(x^2) = int e^t dt/2 + c `
= `1/2 int e^t dt+ c `
= `e^t/2 + c`
∴ `y e ^(x^2) = 1/2 e^(x^2) + 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 |
|
\[x\frac{dy}{dx} + y = y^2\]
|
\[y = \frac{a}{x + a}\]
|
Differential equation \[\frac{d^2 y}{d x^2} - y = 0, y \left( 0 \right) = 2, y' \left( 0 \right) = 0\] Function y = ex + e−x
(1 + x2) dy = xy dx
xy (y + 1) dy = (x2 + 1) dx
tan y \[\frac{dy}{dx}\] = sin (x + y) + sin (x − y)
Solve the following differential equation:
\[xy\frac{dy}{dx} = 1 + x + y + xy\]
Find the particular solution of the differential equation \[\frac{dy}{dx} = - 4x y^2\] given that y = 1, when x = 0.
Solve the following initial value problem:
\[\frac{dy}{dx} + y \cot x = 4x\text{ cosec }x, y\left( \frac{\pi}{2} \right) = 0\]
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 slope of a curve at each of its points is equal to the square of the abscissa of the point. Find the particular curve through the point (−1, 1).
Write the differential equation obtained eliminating the arbitrary constant C in the equation xy = C2.
The integrating factor of the differential equation \[\left( 1 - y^2 \right)\frac{dx}{dy} + yx = ay\left( - 1 < y < 1 \right)\] is ______.
y2 dx + (x2 − xy + y2) dy = 0
Form the differential equation representing the family of curves y = a sin (x + b), where a, b are arbitrary constant.
y2 dx + (xy + x2)dy = 0
Select and write the correct alternative from the given option for the question
The differential equation of y = Ae5x + Be–5x is
Solve the following differential equation y log y = `(log y - x) ("d"y)/("d"x)`
Solve the following differential equation
`y log y ("d"x)/("d"y) + x` = log y
The differential equation of all non horizontal lines in a plane is `("d"^2x)/("d"y^2)` = 0
lf the straight lines `ax + by + p` = 0 and `x cos alpha + y sin alpha = p` are inclined at an angle π/4 and concurrent with the straight line `x sin alpha - y cos alpha` = 0, then the value of `a^2 + b^2` is
