Solve x(dy/dx)+x²=5 with initial condition y=2.5 at x=1

[mathi85]

Hi!

I would like to ask anyone with some spare time to check my assignment questions. Last time I was asked to post one task at a time so I will.
Thank you in advance for your time.

Task 1:

Solve the differential equation: x(dy/dx)+x²=5
given that y=2.5 when x=1

Solution:

x(dy/dx)+x²=5
x(dy/dx)=5-x² /:x
(dy/dx)=(5-x²)/(x)
∴y=∫ 5/x-x dx = 5lnx-(1/2)x²+c

General Solution:

y=5lnx-(1/2)x²+c
∴2.5=5ln(1)-(1/2)(1)²+c
∴c=3

Particular Solution:

y=5lnx-(1/2)x²+3

 

[vela]

It looks okay. I didn't see anything obviously wrong.

You can always check the answer yourself, and this is a habit you should get into. Plug it back into the original differential equation to see if it satisfies the equation. If it does and it meets the initial condition, you're done.

 

[mathi85]

Thank you Vela!