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=R*3.6 Show that an ODE is exact or can be made exact by IFM, find h=

Show That
is exact, or can be made exact by IFM. Then find $$h$$

Solution
On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.

Verifying for Exactness
Equation ($$) can be shown to have components $$N(x,y)$$ and $$M(x,y)$$ as follows: This corresponds with the First Exactness condition:

However, the Second Exactness Condition: is not met, because, and, so,

Making the ODE exact, using IFM, and finding h
Considering the general formula ($$) from the previous question: We can deduce that the individual components of equation ($$) are as follows: $$ c(y)=y^4 $$ $$b(y)=0$$ $$\bar b(x) = \frac{1}{3}x^3$$ $$a(x)=5x^3+2$$ $$c(x)=sin(x)$$ $$\bar c(y) = \frac{1}{5}y^5 $$ where $$b(x)$$ and $$c(y)$$ can be found through derivation, $$ b(x) =\frac {d(\frac{1}{3}x^3)}{dx} = x^2$$ $$c(y)=\frac {d(\frac{1}{5}y^5)}{dy} = y^4 $$ Using the following definitions, we can find $$h$$, the integrating factor: and, where, $$N$$, $$B_x$$ and $$M_y$$ are defined as: Substituting into equation ($$), we get: Expanding, and now, by using the following definition of $$h(x)$$, the integrating factor, and converting $$n(x)$$ to $$n(s)$$, Substituting in for $$n(x)=n(s)$$, where $$k=d$$ and $$s$$ is converted back to $$x$$ the final solution gives:

Given
$$a(x)=sin(x^3)$$ $$b(x)=cos(x)$$ $$c(y)=exp(2y)$$

Create an N2-ODE using the above values, and verify for exactness. If it is not exact, use IFM to make it exact. Furthermore, integrate the final value for $$\phi (x,y)$$

Solution
On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.

Creating the N2-ODE
To create an N2-ODE, we can use equation ($$) and solving for the missing values: Resulting in the following N2-ODE, Since the equation is equal to zero, the first exactness condition is met. To verify the second, we must satisfy Equation ($$) and, so, Hence, we must find an integrating factor to make this N2-ODE exact.

Making the ODE exact, using IFM to find h
Following equations ($$) through ($$), we get: We can then substitute into Equation ($$)

Integrating phi
Now we can multiply ($$) and ($$), while canceling out $$exp(2y)$$ to get, In this case, After the derivation of  $$ \frac {\partial \bar M (x,y)}{\partial x}$$, our final solution is: so,