The integrating factor of x(dy/dx)+(1+x)y=x is

Question

The integrating factor of x(dy/dx)+(1+x)y=x is (A)  x  (B)  2x  (C)  ex log x  (D)  ex

Tardigrade Admin · Accepted Answer

Given differential equation is x(dy/dx)+(1+x)y=x ⇒ (dy/dx)+( (1+x/x) )y=1 Which is linear differential equation Hence, IF=e∫(1+x/x)dx=e∫( (1/x)+1 )dx =e log x+x =e log x.ex =xex

Q. The integrating factor of $x \frac{d y}{d x} + (1 + x) y = x$ is

$x$

$2 x$

$e^{x l o g x}$

$e^{x}$

$x e^{x}$

Given differential equation is $x \frac{d y}{d x} + (1 + x) y = x$
$\Rightarrow$ $\frac{d y}{d x} + (\frac{1 + x}{x}) y = 1$
Which is linear differential equation Hence, $I F = e^{\int \frac{1 + x}{x} d x} = e^{\int (\frac{1}{x} + 1) d x}$
$= e^{l o g x + x}$
$= e^{l o g x} . e^{x}$
$= x e^{x}$

Q. The integrating factor of xdxdy​+(1+x)y=x is

x

2x

exlogx

ex

xex

Given differential equation is xdxdy​+(1+x)y=x ⇒ dxdy​+(x1+x​)y=1 Which is linear differential equation Hence, IF=e∫x1+x​dx=e∫(x1​+1)dx =elogx+x =elogx.ex =xex