Find the differential equation of the family of curves y=e^mx , where m is an arbitrary constant.

Answers (1)

Solution:     Given that    ,      y=e^mx    .......(1)

Solution:  Given ,

                  fracx^2a^2+fracy^2a^2+w=1   with  w as parameter.     ......(1)

             Diffrentiating (1) ,  w.r.t  x ,

            Rightarrow     frac2xa^2+frac2ya^2+wfracmathrmd ymathrmd x=0Rightarrow frac1a^2+w=-fracxa^2yfracmathrmd xmathrmd y   ......(2)

         Eliminating  w  fram (1) and (2) , we get

           Rightarrow   fracx^2a^2+y^2(-fracxa^2yfracmathrmd xmathrmd y)=1      or  fracx^2a^2-fracxya^2frac1fracmathrmd ymathrmd x=1,  ......(3)

       Which is the diffrential equation of the given family of curve (1).

       Replacing  fracmathrmd ymathrmd x   by -fracmathrmd xmathrmd y  in (3)  the differential equation of the orthiogonal trajectories is

      Rightarrow       fracx^2a^2-fracxya^2(-frac1fracmathrmd xmathrmd y)=1   or    fracx^2a^2-fracxya^2fracmathrmd ymathrmd x=1

     Rightarrow      fracxya^2fracmathrmd ymathrmd x=fraca^2-x^2a^2       or     ydy=[fraca^2x-x]dx

    Integrating ,     fracy^22=a^2log x-fracx^22+fracc2Rightarrow x^2+y^2-2a^2log x=c.

        @ Anagh Sharma

              Diffrentiating (1)  w.r.t  x , we get

               Rightarrow            fracmathrmd ymathrmd x=me^mx=my         ......(2)

              Rightarrow           m=frac1yfracmathrmd ymathrmd x.        ......(3)

            Now from (1) and (2)   Rightarrow  mx=log_eyRightarrow m= fraclog_eyx   .....(4)

             Eliminating  m from (3) and (4) , we get

          Rightarrow       frac1yfracmathrmd ymathrmd x=frac1xlog_ey.

Most Viewed Questions

Preparation Products

Knockout NEET 2024

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 40000/-
Buy Now
Knockout NEET 2025

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 45000/-
Buy Now
NEET Foundation + Knockout NEET 2024

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 54999/- ₹ 42499/-
Buy Now
NEET Foundation + Knockout NEET 2024 (Easy Installment)

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 3999/-
Buy Now
NEET Foundation + Knockout NEET 2025 (Easy Installment)

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Unlimited Mock Tests and Personalized Analysis Reports, 24x7 Doubt Chat Support,.

₹ 3999/-
Buy Now
Boost your Preparation for JEE Main with our Foundation Course
 
Exams
Articles
Questions