Advertisements
Advertisements
Question
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
Solution
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
RELATED QUESTIONS
Form the differential equation of the family of hyperbolas having foci on x-axis and centre at the origin.
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 y = cx + 2c2 is a solution of the differential equation
For the following differential equation verify that the accompanying function is a solution:
| Differential equation | Function |
|
\[y = \left( \frac{dy}{dx} \right)^2\]
|
\[y = \frac{1}{4} \left( x \pm a \right)^2\]
|
Differential equation \[\frac{d^2 y}{d x^2} - \frac{dy}{dx} = 0, y \left( 0 \right) = 2, y'\left( 0 \right) = 1\]
Function y = ex + 1
(ey + 1) cos x dx + ey sin x dy = 0
(y2 + 1) dx − (x2 + 1) dy = 0
Solve the following differential equation:
(xy2 + 2x) dx + (x2 y + 2y) dy = 0
Solve the following differential equation:
\[xy\frac{dy}{dx} = 1 + x + y + xy\]
Find the particular solution of edy/dx = x + 1, given that y = 3, when x = 0.
Solve the following initial value problem:-
\[x\frac{dy}{dx} - y = \log x, y\left( 1 \right) = 0\]
Solve the following initial value problem:-
\[\frac{dy}{dx} + y \tan x = 2x + x^2 \tan x, y\left( 0 \right) = 1\]
Solve the following initial value problem:-
\[\frac{dy}{dx} + 2y \tan x = \sin x; y = 0\text{ when }x = \frac{\pi}{3}\]
Solve the following initial value problem:-
\[\tan x\left( \frac{dy}{dx} \right) = 2x\tan x + x^2 - y; \tan x \neq 0\] given that y = 0 when \[x = \frac{\pi}{2}\]
Find the equation of the curve such that the portion of the x-axis cut off between the origin and the tangent at a point is twice the abscissa and which passes through the point (1, 2).
Show that all curves for which the slope at any point (x, y) on it is \[\frac{x^2 + y^2}{2xy}\] are rectangular hyperbola.
Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y \sin x = 1\], is
Choose the correct alternative.
The differential equation of y = `k_1 + k_2/x` is
A solution of a differential equation which can be obtained from the general solution by giving particular values to the arbitrary constants is called ___________ solution.
State whether the following is True or False:
The degree of a differential equation is the power of the highest ordered derivative when all the derivatives are made free from negative and/or fractional indices if any.
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
Given that `"dy"/"dx"` = yex and x = 0, y = e. Find the value of y when x = 1.
Integrating factor of the differential equation `x "dy"/"dx" - y` = sinx is ______.
The value of `dy/dx` if y = |x – 1| + |x – 4| at x = 3 is ______.
