idku one year ago my first Bernoulli DE ... (just looked at it on lamar tutorial and will see how will I do this one)

1. idku

$$\large \color{black}{y'=4y+e^xy^3}$$

2. idku

(this example is completely made up, not on the tutorial)

3. amistre64

how do you know it satisifies a Berny then?

4. amistre64

im thinking (couild be misremembering) that a Berny is a reduction process

5. idku

$$\large \color{black}{y'=4y+e^xy^{3}}$$ $$\large \color{black}{y'-4y=e^xy^{3}}$$ $$\large \color{black}{y'y^{-3}-4y^{-2}=e^x}$$ and then I substitute, $$\large \color{black}{v=y^{-2}}$$ $$\large \color{black}{v'=-2y'y^{-3}}$$ ---> $$\large \color{black}{(-1/2)v'=y'y^{-3}}$$ my DE becomes, $$\large \color{black}{\dfrac{-1}{2}v'-4v=e^x}$$ For convenience, $$\large \color{black}{v'+8v=2e^x}$$ Integrating factor: $$\large \color{black}{e^{4v^2}}$$ $$\large \color{black}{v'e^{4v^2}+e^{4v^2}8v=e^{4v^2}2e^x}$$ $$\large \color{black}{dv/dx(ve^{4v^2})=e^{4v^2}2e^x}$$ wait I am getting stuck here a bit

6. idku

how would I then integrate both sides, if I have that on the RIGHT side?

7. idku

oh, I missed a minus on the right side (when I multiplied by -2)

8. idku

$$\large \color{black}{dv/dx(ve^{4v^2})=-e^{4v^2}2e^x}$$

9. idku

the integrating facto is wrong? is it just e^x because integrating factor is with wespect to x, not v?

10. amistre64

v e^(8v) = v' e^(8v) + 8v e^(8v)

11. amistre64

hmm, dont think thats quite right either

12. idku

$$\large \color{black}{v'+8v=-2e^x}$$ $$\large \color{black}{v'e^{8x}+e^{8x}8v=-2e^{8x}e^x}$$

13. idku

like that?

14. amistre64

that seems better yes

15. idku

$$\large \color{black}{ve^{8x}=\int -2e^{9x}dx}$$ $$\large \color{black}{ve^{8x}= (-2/9)e^{9x}+c}$$ $$\large \color{black}{v= (-2/9)e^{x}+c/e^{8x}}$$

16. idku

Like this?

17. amistre64

so far so good

18. idku

$$\large \color{black}{v= (-2/9)e^{x}+C/e^{8x}}$$ $$\large \color{black}{y^{-2}= (-2/9)e^{x}+C/e^{8x}}$$ and then, $$\large \color{black}{y^2=\dfrac{1}{ (-2/9)e^{x}+C/e^{8x}}}$$ $$\large \color{black}{y=\pm\sqrt{\dfrac{1}{ (-2/9)e^{x}+C/e^{8x}}}}$$ and the rest is a matter of algebraic manipulation if I have done it correclty.

19. amistre64

the process seems fair

20. idku

Awesome, I prevailed it! Thank you amistre for your confirmation!

21. amistre64

youre welcome, i didnt scrutinize it, but it seemed to be in good form overall.

22. idku

Yes, fine for the first time. (Especially considering the fact that I am not really a "math guy")

23. idku

Thank you once again.