Linear Problem - Slope of Curve Through Point

Hi all,

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)

or

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?

Thanks

John

re: Linear Problem - Slope of Curve Through Point

Quote:

Originally Posted by

**celtic1234** Hi all,

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;

NO!! but you cannot integrate y with respect to **x**! y is an unknown function of x.

Instead, you need to **separate** the x and y terms:

Now integrate

Quote:

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)

or

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?

Thanks

John

No, your method is not correct. In particular, you cannot integrate y, an unknown function of x, with respect to x. is NOT the same as .

Re: Linear Problem - Slope of Curve Through Point

ok,

so if i am getting what you re saying;

integrate both sides:

At point (1,-2)

Substituting into the solution for C

Is this it?

John