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
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
Solve the following initial value problem:-
\[dy = \cos x\left( 2 - y\text{ cosec }x \right)dx\]
The rate of increase in the number of bacteria in a certain bacteria culture is proportional to the number present. Given the number triples in 5 hrs, find how many bacteria will be present after 10 hours. Also find the time necessary for the number of bacteria to be 10 times the number of initial present.
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.
Write the differential equation obtained by eliminating the arbitrary constant C in the equation x2 − y2 = C2.
The differential equation satisfied by ax2 + by2 = 1 is
The differential equation \[x\frac{dy}{dx} - y = x^2\], has the general solution
In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-
y = ex + 1 y'' − y' = 0
Determine the order and degree of the following differential equations.
| Solution | D.E. |
| y = 1 − logx | `x^2(d^2y)/dx^2 = 1` |
Find the differential equation whose general solution is
x3 + y3 = 35ax.
Solve the following differential equation.
y dx + (x - y2 ) dy = 0
Choose the correct alternative.
The integrating factor of `dy/dx - y = e^x `is ex, then its solution is
Solve the differential equation:
dr = a r dθ − θ dr
The solution of differential equation `x^2 ("d"^2y)/("d"x^2)` = 1 is ______
Solve the following differential equation `("d"y)/("d"x)` = x2y + y
Solve the differential equation `"dy"/"dx" + 2xy` = y
Solve: `("d"y)/("d"x) = cos(x + y) + sin(x + y)`. [Hint: Substitute x + y = z]
Solution of `x("d"y)/("d"x) = y + x tan y/x` is `sin(y/x)` = cx
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
In mathematics and science, what primary purpose do differential equations serve?
