Mathematics

# $\frac{dy}{dx}=xy+x+y+1$

##### SOLUTION
Given $\dfrac{d y}{d x}=x y+x+y+1$
$\dfrac{d y}{d x}=x(y+1)+(y+1)$
$\dfrac{d y}{d x}=(x+1)(y+1)$
$\implies \dfrac{d y}{y+1}=(x+1) d x$
Integrating on both sides
$\implies \displaystyle\int \dfrac{d y}{y+1}=\int (x+1)d x$
$\implies \ln (y+1)=\dfrac{x^2}{2}+x+C$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 Single Correct Medium
$\displaystyle \int \frac{\sin x+\cos x}{\sqrt{\left ( 1+\sin 2x \right )}}$dx is
• A. $\displaystyle \sin x + C$
• B. $\displaystyle \cos x+C$
• C. $\displaystyle \tan x+C$
• D. $x+C$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle \int \dfrac{x-\sin{x}}{1-\cos{c}}dx$
• A. $-x\cot{\dfrac{x}{2}}+c$
• B. $\cot{\dfrac{x}{2}}+c$
• C. $None \ of\ these$
• D. $x\cot{\dfrac{x}{2}}+c$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
If $\displaystyle\int { { x }^{ 13/2 }.{ \left( 1+{ x }^{ 5/2 } \right) }^{ 1/2 }dx } =P{ \left( 1+{ x }^{ 5/2 } \right) }^{ 7/2 }+Q{ \left( 1+{ x }^{ 5/2 } \right) }^{ 5/2 }+R{ \left( 1+{ x }^{ 5/2 } \right) }^{ 3/2 }+C$, then $P,\ Q$ and $R$ are
• A. $P=\frac { 4 }{ 35 } ,\ Q=\frac { 8 }{ 25 } ,\ R=\frac { 4 }{ 15 }$
• B. $P=-\frac { 4 }{ 35 } ,\ Q=-\frac { 8 }{ 25 } ,\ R=\frac { 4 }{ 15 }$
• C. $P=\frac { 4 }{ 35 } ,\ Q=-\frac { 8 }{ 25 } ,\ R=-\frac { 4 }{ 15 }$
• D. $P=\frac { 4 }{ 35 } ,\ Q=-\frac { 8 }{ 25 } ,\ R=\frac { 4 }{ 15 }$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Hard
Solve:
$\displaystyle \int \dfrac{2\cos x}{(1-\sin x)(1+\sin ^2x)}dx$

Let $n \space\epsilon \space N$ & the A.M., G.M., H.M. & the root mean square of $n$ numbers $2n+1, 2n+2, ...,$ up to $n^{th}$ number are $A_{n}$, $G_{n}$, $H_{n}$ and $R_{n}$ respectively.