Advertisements
Advertisements
प्रश्न
Define a differential equation.
उत्तर
Differential equation:
An equation containing an independent variable, a dependent variable and differential coefficients of the dependent variable with respect to the independent variable is called a differential equation.
for example: \[\frac{dy}{dx} = e^{x + y}\]
APPEARS IN
संबंधित प्रश्न
Show that the function y = A cos 2x − B sin 2x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + 4y = 0\].
Verify that y = log \[\left( x + \sqrt{x^2 + a^2} \right)^2\] satisfies the differential equation \[\left( a^2 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = 0\]
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 \[\frac{d^2 y}{d x^2} - \frac{dy}{dx} = 0, y \left( 0 \right) = 2, y'\left( 0 \right) = 1\]
Function y = ex + 1
(sin x + cos x) dy + (cos x − sin x) dx = 0
C' (x) = 2 + 0.15 x ; C(0) = 100
x cos2 y dx = y cos2 x dy
(y + xy) dx + (x − xy2) dy = 0
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:-
\[y' + y = e^x , y\left( 0 \right) = \frac{1}{2}\]
Solve the following initial value problem:-
\[\frac{dy}{dx} + y \tan x = 2x + x^2 \tan x, y\left( 0 \right) = 1\]
Radium decomposes at a rate proportional to the quantity of radium present. It is found that in 25 years, approximately 1.1% of a certain quantity of radium has decomposed. Determine approximately how long it will take for one-half of the original amount of radium to decompose?
The slope of the tangent at each point of a curve is equal to the sum of the coordinates of the point. Find the curve that passes through the origin.
The differential equation \[x\frac{dy}{dx} - y = x^2\], has the general solution
Integrating factor of the differential equation cos \[x\frac{dy}{dx} + y \sin x = 1\], is
Solve the following differential equation : \[y^2 dx + \left( x^2 - xy + y^2 \right)dy = 0\] .
Show that y = ae2x + be−x is a solution of the differential equation \[\frac{d^2 y}{d x^2} - \frac{dy}{dx} - 2y = 0\]
Verify that the function y = e−3x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + \frac{dy}{dx} - 6y = 0.\]
Find the particular solution of the differential equation `"dy"/"dx" = "xy"/("x"^2+"y"^2),`given that y = 1 when x = 0
In the following example, verify that the given function is a solution of the corresponding differential equation.
Solution | D.E. |
y = xn | `x^2(d^2y)/dx^2 - n xx (xdy)/dx + ny =0` |
Determine the order and degree of the following differential equations.
Solution | D.E |
y = aex + be−x | `(d^2y)/dx^2= 1` |
For each of the following differential equations find the particular solution.
(x − y2 x) dx − (y + x2 y) dy = 0, when x = 2, y = 0
Solve the following differential equation.
`dy /dx +(x-2 y)/ (2x- y)= 0`
Choose the correct alternative.
Bacteria increases at the rate proportional to the number present. If the original number M doubles in 3 hours, then the number of bacteria will be 4M in
Choose the correct alternative.
The integrating factor of `dy/dx - y = e^x `is ex, then its solution is
Solve the differential equation:
`e^(dy/dx) = x`
Solve the following differential equation `("d"y)/("d"x)` = x2y + y
Solve the differential equation `"dy"/"dx"` = 1 + x + y2 + xy2, when y = 0, x = 0.
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?