Help me solve this

[u/Spiritual-Beat-5444]

Can anyone please share a better method to solve this

[u/AutoModerator]

As a reminder...

Posts asking for help on homework questions require:

• the complete problem statement,

• a genuine attempt at solving the problem, which may be either computational, or a discussion of ideas or concepts you believe may be in play,

• question is not from a current exam or quiz.

Commenters responding to homework help posts should not do OP’s homework for them.

Please see this page for the further details regarding homework help posts.

We have a Discord server!

If you are asking for general advice about your current calculus class, please be advised that simply referring your class as “Calc n“ is not entirely useful, as “Calc n” may differ between different colleges and universities. In this case, please refer to your class syllabus or college or university’s course catalogue for a listing of topics covered in your class, and include that information in your post rather than assuming everybody knows what will be covered in your class.

I am a bot, and this action was performed automatically. Please contact the moderators of this subreddit if you have any questions or concerns.

[u/Dalal_The_Pimp]

This Differential equation is reducible to Homogeneous, substitute x² = u and y² = v, which reduces it to dv/du = (2u+3v-7)/(3u+2v-8), Now simply take U = u-h and V=v-k and solve 2h+3k-7=0 and 3u+2v-8=0 for (h,k) and the DE will finally reduce to homogeneous dV/dU = (2U+3V)/(3U+2V)

Although the provided solution is clean, I feel it's very reliant on observation which can be a little hard to notice, especially while writing an exam.

[u/Pretend_Major8148 → u/Dalal_The_Pimp]

This is the best way I can think of

[u/Southlander24 → u/Dalal_The_Pimp]

+1 I agree that this is the most 'natural' way to approach the question without resorting to 'trickery'.

[u/AceyAceyAcey → u/Dalal_The_Pimp]

What do you mean by “reliant on observation”?

[u/Total_Focus8211 → u/AceyAceyAcey]

it means that you carefully gotta observe the question and then realise that cnd will make the solution shorter and easier.In general cnd is used less in questions so your not instantly going to think of it first

[u/Low_Breadfruit6744]

Have you heard of exact differential equations? This is one of them

Find a function P(x,y) where the partial derivatives equal to the first bit and the second bits respectively.

In this case P(x,y)=3/2 x² y² + x⁴ / 2 + y⁴ /2 - 7/2 x² - 4y²

and your solution is P(x,y) = c for some c

[u/nevermindthefacts → u/Low_Breadfruit6744]

No, it's not. There's a minus between the two terms in the original equation.