I have started on differential equations today and need some help (as usual).
From what i have read so far.
The solution to a differential equation will be another function or differential equation.
For example the slope of a curve is a differential equation. The solution of this differential equation will actually give you the curve of the function or the equation of the function itself.
Now i have this problem to solve.
"find the curve whose slope at any point (x,y) is equal to 5y and which passes through the point (1,-2)".
Now the slope is given as 5y;
Now this is a differential equation in itself and the solution to the differential equation will give me the curve of the function itself.
rearranging the slope term
multiply across by dx:
To solve the differential equation i need to integrate both sides;
rearranging similar terms:
Now we are given the point (1,-2) so substitute into equation (1) for x and y
Now the solution to the problem is C inserted into equation (1)
Now to check this if i substitute the solution back into the original equation for y and if i differentiate y with respect to x i should get the slope of 5y?
I think i understand it-some of the text overcomplicates this if you ask me(or more probably i have not got the brains to understand it),
can anybody give me a more basic explanation if i have not got it correct,
Is my method/solution correct?