# Linear first order differential equations

• Feb 17th 2011, 03:57 PM
Mcoolta
Linear first order differential equations
I need to solve the equation:

x (dy/dx) + 5y = 7x where y(1) = 7/2

The correct answer i have recieved is y = 7/6 (x + 2x^-5)

How do i get to this answer?
Ive been given this as a hint:

x (dy/dx) + ay = bx^n can be simplified by multiplying both sides by x^(a-1)

To give x^a (dy/dx) + a(x^(a-1))y = bx^(n+a-1)

Which can be written as d/dx((x^a)y)

But this doesnt seem to help me? Thanks
• Feb 17th 2011, 04:14 PM
pickslides
I would divide both sides by $\displaystyle \displaystyle x$ giving

$\displaystyle \displaystyle y' +\frac{5y}{x}=7$

Now find an integrating factor $\displaystyle \displaystyle e^{\int \frac{5}{x}~dx}$

Multiply this guy through the equation and everything should start to piece together...
• Feb 18th 2011, 01:43 AM
Ackbeet
The equation is also Cauchy-Euler. You could assume a solution of the form $\displaystyle y=x^{m},$ and plug it into the DE to solve for $\displaystyle m.$

Cheers.
• Feb 18th 2011, 04:35 AM
topsquark
It's also separable.
$\displaystyle \displaystyle x~\frac{dy}{dx} + 5y = 7x \implies \frac{dy}{7 - 5y} = \frac{dx}{x}$

-Dan
• Feb 18th 2011, 04:40 AM
Ackbeet
Reply to topsquark: I think you dropped the x multiplying the 7, which might change your result.