Answer
451.2k+ views
Hint: First of all in any differential equation try to separate the variable with their respective differential and then start further solving by method of variable separable form of differential equation, otherwise think of another method that you already know to solve it.
Complete step-by-step answer:
We will separate the variable and try to solve it so, let’s begin with that and we will get;
\[\Rightarrow \dfrac{y}{y+1}dy=\dfrac{{{e}^{x}}}{{{e}^{x}}+1}dx\]
Now, further we will integrate both sides and we get;
\[\Rightarrow \int{\dfrac{y}{y+1}}dy=\int{\dfrac{{{e}^{x}}}{{{e}^{x}}+1}dx}\]
\[\Rightarrow \int{1-\dfrac{1}{1+y}}dy=\int{\dfrac{{{e}^{x}}}{{{e}^{x}}+1}}dx\].................................(a)
Now, we will integrate the both sides of the equality separately as shown below;
\[\Rightarrow \int{1-\dfrac{1}{y+1}}dy=y-\ln \left| y+1 \right|\text{ + }\ln c\text{ }......................\text{(1)}\]
Also, we have $\int{\dfrac{{{e}^{x}}}{{{e}^{x}}+1}dx}$.
Here, let’s suppose that $1+{{e}^{x}}=u$.
$\Rightarrow {{e}^{x}}dx=du$.
Now, we will substitute these values in above integral and we get :
$\Rightarrow \int{\dfrac{1}{u}du=\ln u=\ln (1+{{e}^{x}})...................(2)}$
Now, we will substitute the above value of integrals (1) and (2) in the equation (a) and we get;
\[\begin{align}
& y-\ln \left| y+1 \right|+\ln c=\ln (1+{{e}^{x}}) \\
& \Rightarrow y=\ln (1+{{e}^{x}})+\ln \left| y+1 \right|-\ln c \\
\end{align}\]
Here, we will use the logarithmic formulae which are given below.
These formulae are \[\ln a+\ln b=\ln (ab)\text{ }\]and $\ln a-\ln b=\ln (\dfrac{a}{b})$.
So, on further simplification we will get,
\[\begin{align}
& \Rightarrow y=\ln \dfrac{(1+{{e}^{x}})\left| 1+y \right|}{c} \\
& \Rightarrow {{e}^{y}}=\dfrac{(1+{{e}^{x}})\left| 1+y \right|}{c} \\
& \Rightarrow c{{e}^{y}}=(1+{{e}^{x}})\left| 1+y \right| \\
& \Rightarrow \pm c{{e}^{y}}=(1+{{e}^{x}})(1+y) \\
\end{align}\]
Hence, the solution of the given differential equation is \[({{e}^{x}}+1)(1+y)=\pm c{{e}^{y}}\].
Therefore, the correct option of the above question will be C.
NOTE:
Be careful while doing calculations because there are many places where you can make a mistake. Take care of signs during calculation since it will also change your final answer.
Also take care of the modulus sign which is very important and if you will miss it then the whole solution will become wrong and you will get the incorrect option.
Complete step-by-step answer:
We will separate the variable and try to solve it so, let’s begin with that and we will get;
\[\Rightarrow \dfrac{y}{y+1}dy=\dfrac{{{e}^{x}}}{{{e}^{x}}+1}dx\]
Now, further we will integrate both sides and we get;
\[\Rightarrow \int{\dfrac{y}{y+1}}dy=\int{\dfrac{{{e}^{x}}}{{{e}^{x}}+1}dx}\]
\[\Rightarrow \int{1-\dfrac{1}{1+y}}dy=\int{\dfrac{{{e}^{x}}}{{{e}^{x}}+1}}dx\].................................(a)
Now, we will integrate the both sides of the equality separately as shown below;
\[\Rightarrow \int{1-\dfrac{1}{y+1}}dy=y-\ln \left| y+1 \right|\text{ + }\ln c\text{ }......................\text{(1)}\]
Also, we have $\int{\dfrac{{{e}^{x}}}{{{e}^{x}}+1}dx}$.
Here, let’s suppose that $1+{{e}^{x}}=u$.
$\Rightarrow {{e}^{x}}dx=du$.
Now, we will substitute these values in above integral and we get :
$\Rightarrow \int{\dfrac{1}{u}du=\ln u=\ln (1+{{e}^{x}})...................(2)}$
Now, we will substitute the above value of integrals (1) and (2) in the equation (a) and we get;
\[\begin{align}
& y-\ln \left| y+1 \right|+\ln c=\ln (1+{{e}^{x}}) \\
& \Rightarrow y=\ln (1+{{e}^{x}})+\ln \left| y+1 \right|-\ln c \\
\end{align}\]
Here, we will use the logarithmic formulae which are given below.
These formulae are \[\ln a+\ln b=\ln (ab)\text{ }\]and $\ln a-\ln b=\ln (\dfrac{a}{b})$.
So, on further simplification we will get,
\[\begin{align}
& \Rightarrow y=\ln \dfrac{(1+{{e}^{x}})\left| 1+y \right|}{c} \\
& \Rightarrow {{e}^{y}}=\dfrac{(1+{{e}^{x}})\left| 1+y \right|}{c} \\
& \Rightarrow c{{e}^{y}}=(1+{{e}^{x}})\left| 1+y \right| \\
& \Rightarrow \pm c{{e}^{y}}=(1+{{e}^{x}})(1+y) \\
\end{align}\]
Hence, the solution of the given differential equation is \[({{e}^{x}}+1)(1+y)=\pm c{{e}^{y}}\].
Therefore, the correct option of the above question will be C.
NOTE:
Be careful while doing calculations because there are many places where you can make a mistake. Take care of signs during calculation since it will also change your final answer.
Also take care of the modulus sign which is very important and if you will miss it then the whole solution will become wrong and you will get the incorrect option.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)