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EastWind [94]
3 years ago
15

Solve the equation. StartFraction dx Over dt EndFraction equals StartFraction 1 Over x e Superscript t plus 7 x EndFraction An i

mplicit solution in the form ​F(t,x)equals is nothing equals​C, where C is an arbitrary constant. ​(Type an expression using t and x as the​ variables.)
Mathematics
1 answer:
MAVERICK [17]3 years ago
8 0

Answer:

\frac{1}{49}(7x-1)e^{7x}+e^{-t}=C

Step-by-step explanation:

We are given that

\frac{dx}{dt}=\frac{1}{xe^{t+7x}}

We have to find the implicit function

Using separation variable method

\frac{dx}{dt}=\frac{1}{xe^t\cdot e^{7x}}

By using property x^a\cdot x^y=x^{a+y}

xe^{7x}dx=e^{-t}dt

By using property \frac{1}{x^a}=x^{-a}

Taking integration on both sides

\int xe^{7x}dx=\int e^{-t}dt

Parts integration method

\int u\cdot v dx=u\int vdx-\int (\frac{du}{dx}\int vdx)dx

By parts integration method

x\int e^{7x}dx-\int (\frac{dx}{dx}\int e^{7x}dx)dx=-e^{-t}+C

Using formula \int e^{ax} dx=\frac{e^{ax}}{a}+C

\frac{xe^{7x}}{7}-\frac{1}{7}\int e^{7x}dx=-e^{-t}+C

\frac{xe^{7x}}{7}-\frac{1}{49}e^{7x}+e^{-t}=C

\frac{1}{49}(7x-1)e^{7x}+e^{-t}=C

We are given that

F(x,t)=C

F(x,t)=\frac{1}{49}(7x-1)e^{7x}+e^{-t}=C

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