Mathematics

The solution of $$e^x\sqrt{1-y^2}dx+\dfrac{y}{x}dy=0$$.


SOLUTION
$${ e }^{ x }\sqrt { 1-{ y }^{ 2 } } dx=\cfrac { -y }{ x } dy$$
On separating the variables and integrating, we get
$$\int { { e }^{ x }xdx } =\int { \cfrac { -y }{ \sqrt { 1-{ y }^{ 2 } }  } dy } $$
For RHS, let $$I=\int { \cfrac { -y }{ \sqrt { 1-{ y }^{ 2 } }  } dy } $$
Let $$\sqrt { 1-{ y }^{ 2 } } =z$$
So, $$\cfrac { -2y }{ \sqrt { 1-{ y }^{ 2 } }  } dy=dz$$
So, $$\cfrac { -y }{ \sqrt { 1-{ y }^{ 2 } }  } dy=dz$$
So, $$z=\int { dz } =z+c=\sqrt { 1-{ y }^{ 2 } } +c$$
For RHS,
Let $$J=\int { x{ e }^{ x }dx } $$
Using product rule, we get
$$J=x\int { { e }^{ x }dx } -\int { \left[ \cfrac { d\left( x \right)  }{ dx } \int { { e }^{ x }dx }  \right] dx } $$
$$J+x{ e }^{ x }-{ e }^{ x }={ e }^{ x }\left( x-1 \right) +c$$
So, going to original equation, we get
$${ e }^{ x }\left( x-1 \right) =\sqrt { 1-{ y }^{ 2 } } +c$$
View Full Answer

Its FREE, you're just one step away


Subjective Medium Published on 17th 09, 2020
Next Question
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86
Enroll Now For FREE

Realted Questions

Q1 Subjective Hard
Evaluate the following integral:
$$\displaystyle \quad \int { \cfrac { \sin { 2x }  }{ \sin { \left( x-\cfrac { \pi  }{ 6 }  \right)  } \sin { \left( x+\cfrac { \pi  }{ 6 }  \right)  }  }  } dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Single Correct Medium
Solve $$\int _{0}^{\pi} x\tan^{2}{x}dx$$
  • A. $$\dfrac{\pi^{2}}{2}$$
  • B. $$\dfrac{3\pi^{2}}{2}$$
  • C. None of these
  • D. $$-\dfrac{\pi^{2}}{2}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Single Correct Medium
Evaluate : $$\displaystyle \int^{\pi/2}_0\sqrt{1+\cos 2x}dx$$
  • A. $$\dfrac{3}{2}$$
  • B. $$\sqrt{3}$$
  • C. $$2$$
  • D. $$\sqrt{2}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Subjective Medium
Solve $$xdy-ydx=x^3dy+x^2ydx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Subjective Easy
Evaluate:
$$ \int_{}^{} {\frac{{ - 1}}{{\sqrt {1 - {x^2}} }}dx} $$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer