# Thread: general solution of ode

1. ## general solution of ode

Question: find the general solution of xX' = aX
iknow it's kinda simple ode.. but, i just dont know y i cant get the correct answer..

solution..

xX' = aX
x dX/dx = aX
d/dx X = aX/x
X = ∫aX/x dx
X = aX ln |x| + C

the general solution is Ax^a

problem : i cant get the correct answer..

2. Originally Posted by nameck
Question: find the general solution of xX' = aX
iknow it's kinda simple ode.. but, i just dont know y i cant get the correct answer..

solution..

xX' = aX
x dX/dx = aX
d/dx X = aX/x
X = ∫aX/x dx
X = aX ln |x| + C

the general solution is Ax^a

problem : i cant get the correct answer..
I'll use x's and y's - it's a little easier.

$\displaystyle x \frac{dy}{dx} = ay \; \text{(we separate)}\; \frac{dy}{y} = a \frac{dx}{x}$

integrating gives

$\displaystyle \ln y = a \ln x + c \; \text{(let} c = \ln A)$

so

$\displaystyle \ln y = \ln A + a\ln x = \ln Ax^a$
giving

$\displaystyle y = Ax^a.$

3. I'll use x's and y's - it's a little easier.

can i simply change the variable?
bcoz, X is equal tu X(x) function of x..

4. Originally Posted by nameck
I'll use x's and y's - it's a little easier.

can i simply change the variable?
bcoz, X is equal tu X(x) function of x..
Sure. Just keep it straight and remember to switch back in the end.