Differential Equations

Question

Solve:
$straight y space straight e to the power of straight x over straight y end exponent dx space equals open parentheses xe to the power of straight x over straight y end exponent plus straight y squared close parentheses space space dy comma space space space straight y not equal to 0$

Solution

The given differential equation is
   $straight y space straight e to the power of straight x over straight y end exponent space dx space equals space open parentheses xe to the power of straight x over straight y end exponent plus straight y squared close parentheses space dy$
or $dx over dy space equals space fraction numerator xe to the power of begin display style straight x over straight y end style end exponent plus straight y squared over denominator ye to the power of begin display style straight x over straight y end style end exponent end fraction$
or $dx over dy space equals fraction numerator xe to the power of begin display style straight x over straight y end style end exponent plus straight y squared over denominator ye to the power of begin display style straight x over straight y end style end exponent end fraction$
or $dx over dy space equals space straight x over straight y plus straight y over straight e to the power of begin display style straight x over straight y end style end exponent$
Put y = v y so that $dx over dy equals straight v plus straight y dv over dx$
$therefore space space space space space space space space space straight v plus straight y dv over dy space equals space straight v plus ye to the power of negative straight v end exponent therefore space space space space space space space space straight y dv over dy space equals space straight y space straight e to the power of negative straight v end exponent space space space space space space space space space space space or space space space space space space dv over dy space equals straight e to the power of negative straight v end exponent$
Separting the variables and integration, we get,
   $integral straight e to the power of straight v dv space equals space integral dy space space space space space space space space space or space space space space space straight e to the power of straight v space equals space straight y plus straight c$
or $straight e to the power of straight x over straight y end exponent space equals space straight y plus straight c$

For Daily Free Study Material Join wiredfaculty

Differential Equations

Solve:
$straight y space straight e to the power of straight x over straight y end exponent dx space equals open parentheses xe to the power of straight x over straight y end exponent plus straight y squared close parentheses space space dy comma space space space straight y not equal to 0$

Some More Questions From Differential Equations Chapter

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Phone Number

Email Address

Loaction