Differential equation right hand function

[Doubell]

Homework Statement

The question specifies the auxiliary equation given is (D^2 + D - 2) = (e^x)/(x)
the method of variation of parameter must be used to find the particular solution to the right hand function. then finally the general soultion should be stated.

Homework Equations

variation of paremeters formula.
y_p = -y₁∫y₂ * g(x)/ (w(y₁ , y₂))
+ y₂∫y₁ * g(x)/ (w(y₁ , y₂))

The Attempt at a Solution

i had solved the complementary function and had gotten y_c = c₁e^-x + c₂e^-2x. then after applying the formula for variation of parameters i had gotten e^-x*lnx/3 - e^-2x/3 * ∫e^3x / x
i cannot obtain an integral for ∫e^3x / x i don't think it can be done have tried various mathods eg by parts, there are no suitable substitutions is it that there is no integral for the expression? i.e. it cannot be integrated.

 

[LCKurtz]

Homework Statement

The question specifies the auxiliary equation given is (D^2 + D - 2) = (e^x)/(x)
the method of variation of parameter must be used to find the particular solution to the right hand function. then finally the general soultion should be stated.

Homework Equations

variation of paremeters formula.
y_p = -y₁∫y₂ * g(x)/ (w(y1₁ , y₂))
+ y₂∫y₁ * g(x)/ (w(y1₁ , y₂))

The Attempt at a Solution

i had solved the complementary function and had gotten y_c = c₁e^-x + c₂e^-2x. then after applying the formula for variation of parameters i had gotten e^-x*lnx/3 - e^-2x/3 * ∫e^3x / x
i cannot obtain an integral for ∫e^3x / x i don't think it can be done have tried various mathods eg by parts, there are no suitable substitutions is it that there is no integral for the expression? i.e. it cannot be integrated.

I haven't worked through this problem, but you are correct that ##\frac {e^{3x}} x## does not have an elementary antiderivative.

 

[Mark44]

Homework Statement

The question specifies the auxiliary equation given is (D^2 + D - 2) = (e^x)/(x)
the method of variation of parameter must be used to find the particular solution to the right hand function. then finally the general soultion should be stated.

Homework Equations

variation of paremeters formula.
y_p = -y₁∫y₂ * g(x)/ (w(y₁ , y₂))
+ y₂∫y₁ * g(x)/ (w(y₁ , y₂))

The Attempt at a Solution

i had solved the complementary function and had gotten y_c = c₁e^-x + c₂e^-2x.

You have a mistake. What you have as e^-x should be e^+x. The other one is fine.

then after applying the formula for variation of parameters i had gotten e^-x*lnx/3 - e^-2x/3 * ∫e^3x / x
i cannot obtain an integral for ∫e^3x / x i don't think it can be done have tried various mathods eg by parts, there are no suitable substitutions is it that there is no integral for the expression? i.e. it cannot be integrated.

 

[Doubell]

You have a mistake. What you have as e^-x should be e^+x. The other one is fine.

yes it is as u say but the problem still remains the same the∫e^3x/x would still have to be determined

 

[Mark44]

yes it is as u say but the problem still remains the same the∫e^3x/x would still have to be determined

But if you have an incorrect function in your variation of parameters formula, you'll definitely get the wrong answer.

Also, "textspeak" (such as "u" for "you") isn't permitted here at PF.

Please show your work in this formula:

y_p = -y₁∫y₂ * g(x)/ (w(y₁ , y₂))
+ y₂∫y₁ * g(x)/ (w(y₁ , y₂))