Advertisements
Advertisements
Question
Solve
`dy/dx + 2/ x y = x^2`
Advertisements
Solution
`dy/dx + 2/ x y = x^2`
The given equation is of the form
`dy/dx + py = Q`
`where, P = 2/x and Q = x^2`
∴ I.F. =`e^(int^(pdx) = e^(2int^(1/xdx) e = ^(2logx) = e^(logx^2) = x^2`
∴ Solution of the given equation is
`y(I.F.) = int Q(I.F.) dx + c_1`
`y(x^2) = int x^2 xx x^2 dx + c_1`
∴ `x ^2 y = x^4 intdx + c_1`
∴ `x^2 y = x^5/5 + c_1`
∴ 5x2 y = x5 + c …[c = 5c1]
APPEARS IN
RELATED QUESTIONS
Form the differential equation representing the family of ellipses having centre at the origin and foci on x-axis.
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 \[x\frac{dy}{dx} = 1, y\left( 1 \right) = 0\]
Function y = log x
xy dy = (y − 1) (x + 1) dx
Solve the following initial value problem:-
\[\left( 1 + y^2 \right) dx + \left( x - e^{- \tan^{- 1} y} \right) dx = 0, y\left( 0 \right) = 0\]
Write the differential equation obtained eliminating the arbitrary constant C in the equation xy = C2.
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?
If xmyn = (x + y)m+n, prove that \[\frac{dy}{dx} = \frac{y}{x} .\]
The differential equation `y dy/dx + x = 0` represents family of ______.
In the following example, verify that the given function is a solution of the corresponding differential equation.
| Solution | D.E. |
| xy = log y + k | y' (1 - xy) = y2 |
Solve the following differential equation.
`dy /dx +(x-2 y)/ (2x- y)= 0`
Solve the following differential equation.
`x^2 dy/dx = x^2 +xy - y^2`
Solve the differential equation:
dr = a r dθ − θ dr
y2 dx + (xy + x2)dy = 0
Solve the differential equation sec2y tan x dy + sec2x tan y dx = 0
Solve the differential equation (x2 – yx2)dy + (y2 + xy2)dx = 0
Solve the following differential equation
`x^2 ("d"y)/("d"x)` = x2 + xy − y2
Choose the correct alternative:
General solution of `y - x ("d"y)/("d"x)` = 0 is
The solution of differential equation `x^2 ("d"^2y)/("d"x^2)` = 1 is ______
Solve the following differential equation
`y log y ("d"x)/("d"y) + x` = log y
Verify y = `a + b/x` is solution of `x(d^2y)/(dx^2) + 2 (dy)/(dx)` = 0
y = `a + b/x`
`(dy)/(dx) = square`
`(d^2y)/(dx^2) = square`
Consider `x(d^2y)/(dx^2) + 2(dy)/(dx)`
= `x square + 2 square`
= `square`
Hence y = `a + b/x` is solution of `square`
