zagonek34

Answered

2021-12-27

Using variable separable find the complete solution of the given differential equations

$x{y}^{3}dx+(y+1){e}^{-x}dy$

Answer & Explanation

Maria Lopez

Expert

2021-12-28Added 32 answers

Integrate on both sides

Travis Hicks

Expert

2021-12-29Added 29 answers

Step 1

The given equation is:

$x{y}^{3}dx+(y+1){e}^{-x}dy=0$

We can rewrite this as:

$x{y}^{3}dx=-(y+1){e}^{-x}dy$

Divide both sides by$e}^{-x}{y}^{3$

$\Rightarrow x{e}^{x}dx=-(y+1){y}^{-3}dy$

This is variable separable form

Step 2

We can solve the equation by integrating equation we have got in Step 1

$\Rightarrow \int x{e}^{x}dx=\int -(y+1){y}^{-3}dy$

$\Rightarrow x{e}^{x}-\int {e}^{x}.1dx=-\int ({y}^{-2}+{y}^{-3})dy$

$\Rightarrow x{e}^{x}-{e}^{x}=-\frac{{y}^{-2+1}}{-2+1}-\frac{{y}^{-3+1}}{-3+1}+C$

$\Rightarrow {e}^{x}(x-1)=\frac{1}{y}+\frac{1}{2{y}^{2}}+C$

$\Rightarrow {e}^{x}(x-1)=\frac{2y+1}{2{y}^{2}}+C$

Step 3

Answer:${e}^{x}(x-1)=\frac{2y+1}{2{y}^{2}}+C$

The given equation is:

We can rewrite this as:

Divide both sides by

This is variable separable form

Step 2

We can solve the equation by integrating equation we have got in Step 1

Step 3

Answer:

karton

Expert

2022-01-10Added 439 answers

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