Answer :
Answer:
[tex]\frac{1}{49}(7x-1)e^{7x}+e^{-t}=C[/tex]
Step-by-step explanation:
We are given that
[tex]\frac{dx}{dt}=\frac{1}{xe^{t+7x}}[/tex]
We have to find the implicit function
Using separation variable method
[tex]\frac{dx}{dt}=\frac{1}{xe^t\cdot e^{7x}}[/tex]
By using property [tex]x^a\cdot x^y=x^{a+y}[/tex]
[tex]xe^{7x}dx=e^{-t}dt[/tex]
By using property [tex]\frac{1}{x^a}=x^{-a}[/tex]
Taking integration on both sides
[tex]\int xe^{7x}dx=\int e^{-t}dt[/tex]
Parts integration method
[tex]\int u\cdot v dx=u\int vdx-\int (\frac{du}{dx}\int vdx)dx[/tex]
By parts integration method
[tex]x\int e^{7x}dx-\int (\frac{dx}{dx}\int e^{7x}dx)dx=-e^{-t}+C[/tex]
Using formula [tex]\int e^{ax} dx=\frac{e^{ax}}{a}+C[/tex]
[tex]\frac{xe^{7x}}{7}-\frac{1}{7}\int e^{7x}dx=-e^{-t}+C[/tex]
[tex]\frac{xe^{7x}}{7}-\frac{1}{49}e^{7x}+e^{-t}=C[/tex]
[tex]\frac{1}{49}(7x-1)e^{7x}+e^{-t}=C[/tex]
We are given that
[tex]F(x,t)=C[/tex]
[tex]F(x,t)=\frac{1}{49}(7x-1)e^{7x}+e^{-t}=C[/tex]