Juan Hewlett

Answered

2021-12-21

What is the differential equation of the orthogonal trajectories of the family of curves ${x}^{2}-3xy-2{y}^{2}=C$ ?

Answer & Explanation

autormtak0w

Expert

2021-12-22Added 31 answers

Introduction:

This straightforward question based on differential equations could be resolved by applying the fundamental understanding of the subject as

Solution:

Given: ${x}^{2}-3xy-2{y}^{2}=C$

Now differentiating w.r.t. x we get

$2x-3(y+x{y}^{\prime})-2\left(2y\right){y}^{\prime}=0\text{}as\text{}{\left(xy\right)}^{\prime}={x}^{\prime}y+x{y}^{\prime}\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}{\left(C\right)}^{\prime}=0$

$2x-3y-3x{y}^{\prime}-4y{y}^{\prime}=0\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{or}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}2x-3y-{y}^{\prime}(3x+4y)=0$

Therefore $2x-3y-{y}^{\prime}(3x+4y)=0$ is the required differential equation.

Annie Gonzalez

Expert

2021-12-23Added 41 answers

Given,

Differentiating both sides

Replacing

RizerMix

Expert

2021-12-29Added 437 answers

Given family of Curve is

Since

Diff. both side with respect to x, we get

Hence

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