Advertisements
Advertisements
Question
Solve the following differential equation.
`dy/dx + 2xy = x`
Advertisements
Solution
`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
RELATED QUESTIONS
Verify that y = 4 sin 3x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + 9y = 0\]
Verify that y2 = 4a (x + a) is a solution of the differential equations
\[y\left\{ 1 - \left( \frac{dy}{dx} \right)^2 \right\} = 2x\frac{dy}{dx}\]
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
tan y dx + sec2 y tan x dy = 0
(1 + x) (1 + y2) dx + (1 + y) (1 + x2) dy = 0
Solve the following differential equation:
\[y\left( 1 - x^2 \right)\frac{dy}{dx} = x\left( 1 + y^2 \right)\]
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).
\[x^2 \frac{dy}{dx} = x^2 + xy + y^2 \]
The population of a city increases at a rate proportional to the number of inhabitants present at any time t. If the population of the city was 200000 in 1990 and 250000 in 2000, what will be the population in 2010?
If the marginal cost of manufacturing a certain item is given by C' (x) = \[\frac{dC}{dx}\] = 2 + 0.15 x. Find the total cost function C (x), given that C (0) = 100.
Find the equation of the curve passing through the point \[\left( 1, \frac{\pi}{4} \right)\] and tangent at any point of which makes an angle tan−1 \[\left( \frac{y}{x} - \cos^2 \frac{y}{x} \right)\] with x-axis.
The differential equation
\[\frac{dy}{dx} + Py = Q y^n , n > 2\] can be reduced to linear form by substituting
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
Solve the following differential equation.
y2 dx + (xy + x2 ) dy = 0
Solve the following differential equation
`yx ("d"y)/("d"x)` = x2 + 2y2
Given that `"dy"/"dx" = "e"^-2x` and y = 0 when x = 5. Find the value of x when y = 3.
Solve: `("d"y)/("d"x) = cos(x + y) + sin(x + y)`. [Hint: Substitute x + y = z]
`d/(dx)(tan^-1 (sqrt(1 + x^2) - 1)/x)` is equal to:
