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
Show that the differential equation of which y = 2(x2 − 1) + \[c e^{- x^2}\] is a solution, is \[\frac{dy}{dx} + 2xy = 4 x^3\]
Form the differential equation representing the family of ellipses having centre at the origin and foci on x-axis.
Verify that y = cx + 2c2 is a solution of the differential equation
Show that the differential equation of which \[y = 2\left( x^2 - 1 \right) + c e^{- x^2}\] is a solution is \[\frac{dy}{dx} + 2xy = 4 x^3\]
Differential equation \[x\frac{dy}{dx} = 1, y\left( 1 \right) = 0\]
Function y = log x
Differential equation \[\frac{d^2 y}{d x^2} - y = 0, y \left( 0 \right) = 2, y' \left( 0 \right) = 0\] Function y = ex + e−x
(y2 + 1) dx − (x2 + 1) dy = 0
Solve the following differential equation:
\[y e^\frac{x}{y} dx = \left( x e^\frac{x}{y} + y^2 \right)dy, y \neq 0\]
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).
At every point on a curve the slope is the sum of the abscissa and the product of the ordinate and the abscissa, and the curve passes through (0, 1). Find the equation of the curve.
Find the equation of the curve which passes through the point (1, 2) and the distance between the foot of the ordinate of the point of contact and the point of intersection of the tangent with x-axis is twice the abscissa of the point of contact.
Write the differential equation representing the family of straight lines y = Cx + 5, where C is an arbitrary constant.
The differential equation obtained on eliminating A and B from y = A cos ωt + B sin ωt, is
Which of the following differential equations has y = C1 ex + C2 e−x as the general solution?
Show that y = ae2x + be−x is a solution of the differential equation \[\frac{d^2 y}{d x^2} - \frac{dy}{dx} - 2y = 0\]
Form the differential equation representing the family of parabolas having vertex at origin and axis along positive direction of x-axis.
Solve the following differential equation.
`dy/dx = x^2 y + y`
Solve the following differential equation.
x2y dx − (x3 + y3) dy = 0
Solve the following differential equation `("d"y)/("d"x)` = x2y + y
Solve `x^2 "dy"/"dx" - xy = 1 + cos(y/x)`, x ≠ 0 and x = 1, y = `pi/2`
Solve the differential equation `"dy"/"dx"` = 1 + x + y2 + xy2, when y = 0, x = 0.
