Advertisements
Advertisements
प्रश्न
Determine the order and degree of the following differential equations.
| Solution | D.E. |
| ax2 + by2 = 5 | `xy(d^2y)/dx^2+ x(dy/dx)^2 = y dy/dx` |
Advertisements
उत्तर
ax 2 + by 2 = 5
Differentiating w.r.t. x, we get
`2ax +2by dy/dx = 0` ....(i)
Again, differentiating w.r.t. x, we get
`2a + 2b(dy/dx)^2 + 2by ((d^2y)/dx^2) = 0` ........(ii)
From (i), we get
`a = - (by)/x(dy/dx)`
Substituting the value of a in (ii), we get
`- 2(by)/x(dy/dx) + 2b(dy/dx)^2 + 2by((d^2y)/dx^2) = 0`
∴`- y/x(dy/dx) + (dy/dx)^2 + y((d^2y)/dx^2) = 0`
∴`- y(dy/dx) + x(dy/dx)^2 + xy((d^2y)/dx^2) = 0`
∴`x y((d^2y)/dx^2) + x(dy/dx)^2 =y((dy)/dx) `
∴ Given function is a solution of the given differential equation.
APPEARS IN
संबंधित प्रश्न
Assume that a rain drop evaporates at a rate proportional to its surface area. Form a differential equation involving the rate of change of the radius of the rain drop.
Show that the function y = A cos x + B sin x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + y = 0\]
Verify that y2 = 4ax is a solution of the differential equation y = x \[\frac{dy}{dx} + a\frac{dx}{dy}\]
Show that Ax2 + By2 = 1 is a solution of the differential equation x \[\left\{ y\frac{d^2 y}{d x^2} + \left( \frac{dy}{dx} \right)^2 \right\} = y\frac{dy}{dx}\]
If y(x) is a solution of the different equation \[\left( \frac{2 + \sin x}{1 + y} \right)\frac{dy}{dx} = - \cos x\] and y(0) = 1, then find the value of y(π/2).
Find the particular solution of the differential equation
(1 – y2) (1 + log x) dx + 2xy dy = 0, given that y = 0 when x = 1.
\[x^2 \frac{dy}{dx} = x^2 + xy + y^2 \]
Solve the following initial value problem:
\[\frac{dy}{dx} + y \cot x = 4x\text{ cosec }x, y\left( \frac{\pi}{2} \right) = 0\]
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.
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 \[\frac{du}{dx} + P\left( x \right) u = Q\left( x \right)\]
What is integrating factor of \[\frac{dy}{dx}\] + y sec x = tan x?
y2 dx + (x2 − xy + y2) dy = 0
In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-
`y=sqrt(a^2-x^2)` `x+y(dy/dx)=0`
Form the differential equation representing the family of curves y = a sin (x + b), where a, b are arbitrary constant.
Find the coordinates of the centre, foci and equation of directrix of the hyperbola x2 – 3y2 – 4x = 8.
Solve the following differential equation.
y2 dx + (xy + x2 ) dy = 0
Solve the following differential equation.
(x2 − y2 ) dx + 2xy dy = 0
Select and write the correct alternative from the given option for the question
Differential equation of the function c + 4yx = 0 is
Solve the following differential equation
`yx ("d"y)/("d"x)` = x2 + 2y2
Solve the following differential equation
`y log y ("d"x)/("d"y) + x` = log y
Solve the following differential equation
sec2 x tan y dx + sec2 y tan x dy = 0
Solution: sec2 x tan y dx + sec2 y tan x dy = 0
∴ `(sec^2x)/tanx "d"x + square` = 0
Integrating, we get
`square + int (sec^2y)/tany "d"y` = log c
Each of these integral is of the type
`int ("f'"(x))/("f"(x)) "d"x` = log |f(x)| + log c
∴ the general solution is
`square + log |tan y|` = log c
∴ log |tan x . tan y| = log c
`square`
This is the general solution.
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
The differential equation (1 + y2)x dx – (1 + x2)y dy = 0 represents a family of:
