Problem with ODE

[gino492]

Hi guys, new member here!

I have a problem with what would usually be quite a straight forward linear equation.

I am trying to find the general solution of:

dy/dx +2xy = 4x

Now usually I would just find the integrating factor, multiply both sides to get:

d/dx (exp(x^2)y) = 4xexp(x^2)

Problem. Can't integrate the RHS because of the exp(x^2).

Can I do this by variation of parameters or go in search of another way? Analytically?

Thanks

Gino​

 

[Deleted member 4993]

Hi guys, new member here!

I have a problem with what would usually be quite a straight forward linear equation.

I am trying to find the general solution of:

dy/dx +2xy = 4x

Now usually I would just find the integrating factor, multiply both sides to get:

d/dx (exp(x^2)y) = 4xexp(x^2)

Problem. Can't integrate the RHS because of the exp(x^2).

Can I do this by variation of parameters or go in search of another way? Analytically?

Thanks

Gino​

Why is that?

\(\displaystyle \int 4*x*e^{x^2} dx \)

\(\displaystyle = \ 2\int e^{u} du \)

\(\displaystyle = 2 * e^{x^2} + C\)

 

[Deleted member 4993]

Hi guys, new member here!

I have a problem with what would usually be quite a straight forward linear equation.

I am trying to find the general solution of:

dy/dx +2xy = 4x

Now usually I would just find the integrating factor, multiply both sides to get:

d/dx (exp(x^2)y) = 4xexp(x^2)

Problem. Can't integrate the RHS because of the exp(x^2).

Can I do this by variation of parameters or go in search of another way? Analytically?

Thanks

Gino​

Another way:

y' = 4x - 2xy = 2x(2-y)

dy/(2-y) = 2x dx

ln(C/(2-y)) = x²

C/(2-y) = e^(x²)

y = 2 - C*e^(-x²)

 

[gino492]

Why is that?

\(\displaystyle \int 4*x*e^{x^2} dx \)

\(\displaystyle = \ 2\int e^{u} du \)

\(\displaystyle = 2 * e^{x^2} + C\)

I thought e(x^2) couldn't be integrated straightforwardly, as in it can't be expressed in terms of elementary functions? Maple doesn't like it.

 

[Deleted member 4993]

I thought e(x^2) couldn't be integrated straightforwardly, as in it can't be expressed in terms of elementary functions? Maple doesn't like it.

What you have is NOT \(\displaystyle \int e^{-x^2} dx\) ................ which cannot be integrated without limits.

What you have is \(\displaystyle \int x * e^{-x^2} dx\)................ which can be integrated without limits.