Question

Determine whether there exists an integrating factor that is a function of x only that would make the following differential equations exact. If so, compute the integrating factor and determine the solution to the differential equation.
(7x^2)*(sqrt(x*y))+cos(x*y)dy/dx=0 y(0)=0

Answer 1

I don't think there is an integrating factor that is a function of x only for that differential equation. In my DE book I found a section which says if \[(\delta P/ \delta y - \delta Q/ \delta x)/Q\] is a function of x only, then \[e ^{\int\limits_{}^{}h(x)dx}\] is an integrating factor where \[Pdx + Qdy = 0\] is the form of the differential equation. I calculated this and did not get a function of only x. Check my math but that seems to answer your question.

Answer 2

\[\begin{align} \text{Integrating factor } R =R(x) \\
\\
\qquad&\text{for exactness } \\
\\
\frac{\partial}{\partial y}\big(R(x)P\big) &=\frac{\partial}{\partial x}\big(R(x)Q\big) \\
RP_y &=R'Q+RQ_y \\
R(P_y-Q_x) &=R'Q \\
\frac {R'}R &=\frac{P_y-Q_x}Q \\
\int\frac{d R}R&=\int\frac{P_y-Q_x}Qdx \\
\ln Q &=\int\frac{P_y-Q_x}Qd x \\
R(x) &=\Large e^ {\int\frac{P_y-Q_x}Qd x}
\end{align}\]

But i think your right @JonathanHocker, this only works iff \[\frac{P_y-Q_x}Q=h(x) \] a function of x only. but that dosen't seam to be the case here