Find the equation of given curve in the form ##e^{3y}=f(x)##

[chwala]

Homework Statement

See attached

Relevant Equations

Integration

For part (a);

$$\int e^{3y} \,dy=\int 3x^2\ln x \,dx$$$$\frac{e^{3y}}{3}=x^3\ln x-\frac{x^3}{3}+k$$$$\frac{e^{3}}{3}=e^3-\frac{e^3}{3}+k$$$$\frac{e^{3y}}{3}=x^3\ln x-\frac{x^3}{3}-\frac{e^3}{3}$$$$e^{3y}=3x^3 \ln x-x^3-e^3$$

You may check my working...i do not have the solution.

 

[Mark44]

Homework Statement:: See attached
Relevant Equations:: Integration

View attachment 302689
For part (a);

$$\int e^{3y} \,dy=\int 3x^2\ln x \,dx$$$$\frac{e^{3y}}{3}=x^3\ln x-\frac{x^3}{3}+k$$$$\frac{e^{3}}{3}=e^3-\frac{e^3}{3}+k$$$$\frac{e^{3y}}{3}=x^3\ln x-\frac{x^3}{3}-\frac{e^3}{3}$$$$e^{3y}=3x^3 \ln x-x^3-e^3$$

You may check my working...i do not have the solution.

You can check this for yourself, something you should always do when you solve differential equations. Since this is an initial value problem, checking your solution amounts to doing two things:

1. Verify that the point (e, 1) satisfies the equation $e^{3y}=3x^3 \ln x-x^3-e^3$.
2. Verify that for the implicit equation you found satisfies the DE $\frac{dy}{dx} = \frac{3x^2 \ln(x)}{e^{3y}}$.

 

[Delta2]

I find your work correct. Only thing you could add more detailed steps is in the passage from first to second line, but yes ok it is just one integration by parts.

 

[chwala]

You can check this for yourself, something you should always do when you solve differential equations. Since this is an initial value problem, checking your solution amounts to doing two things:

1. Verify that the point (e, 1) satisfies the equation $e^{3y}=3x^3 \ln x-x^3-e^3$.
2. Verify that for the implicit equation you found satisfies the DE $\frac{dy}{dx} = \frac{3x^2 \ln(x)}{e^{3y}}$.

Thanks @Mark44 ...1. $e^3=3e^3-e^3-e^3$ (thus satisfied).

2.

$3e^{3y}y'=9x^2\ln x +3x^2-3x^2$

$⇒3e^{3y}y'=9x^2\ln x$

$y'=\dfrac{9x^2\ln x}{3e^{3y}}=\dfrac{3x^2\ln x}{e^{3y}}$

 

[Mark44]

Something I've said to many of my students when I taught classes on differential equations. I said it in a previous post here, but it bears repeating.

You can check this for yourself, something you should always do when you solve differential equations.