Clairaut' s Equation x dy/dx = y + log dy/dx * Clairaut' s Equation x dy/dx = y + log dy/dx *
Answers
Answer :
y = cx - logc , where c is any arbitrary constant .
Note :
★ Clairaut's Equation : It is a differential equation of the form y = px + f(p) , where p = dy/dx .
★ The general solution of the Clairaut's Equation can be obtained by replacing p by c , where p = dy/dx and c is any arbitrary constant .
★ Thus , the general solution of the Clairaut's Equation y = px + f(p) is given as y = cx + f(c) , where c is any arbitrary constant .
Solution :
Here ,
The given Clairaut's Equation is ;
x(dy/dx) = y + log(dy/dx)
The given diffential equation can be rewritten as ;
=> xp = y + logp
=> y = xp - logp
=> y = xp + (-logp) , where p = dy/dx
Clearly ,
It is of the form y = px + f(p)
where f(p) = -logp
Thus ,
The general solution of the given Clairaut's Equation will be : y = cx - logc , where c is any arbitrary constant .