Solving differential equations with conditions

Hello, I have a question xy’ = y ; y = cx; y=pi when x = 2. I have to verify that the given function is a solution to the DE. Also find the c that satisfies the condition.

So here I put y = pi and x =2 for the xy' = y.

So i got 2y'= pi which equals to y'= pi/2.

And I found out that y= (pi/2)x + C. So since I am trying to find C in y=Cx, Is the answer (pi/2)??

Thank you.

Re: Solving differential equations with conditions

To verify the given solution $\displaystyle y=cx$ satisfies the given ODE, compute $\displaystyle y'=c$ and now substitute for both into the ODE:

$\displaystyle x\cdot c=cx$

This is true, so the solution is valid.

To find the $\displaystyle c$ that satisfies the given initial condition, simply write:

$\displaystyle y(2)=c\cdot2=\pi\,\therefore\,c=\frac{\pi}{2}$