Question

Solve the given differential equation

Answer 1

Answer:

$\frac{y}{3} + \sqrt{ {x}^{2} + \frac{ {y}^{2} }{9} } = c {x}^{2}$

Step-by-step explanation:

I initially got stuck at the very beginning because I did the product rule wrong T-T

Isolate the y' by dividing out the x in front of it and simplify a bit. Then substitute y/x for a variable (I chose z), replace all applicable variables, and simplify where needed.

At this point it should be separable! Separate the variables as you normally would and integrate to solve.

• For the x portion you could probably, technically leave the +c as is and work with from there. But turning the c into ln(c) [which is fair because the constant is arbitrary] helps us simplify it to be used later on.
• For the z integral, use a trig. substitution for tangent and reduce to get the integral of secant. Remember that you have to swap the variables back from theta into z because the integral is indefinite!

Lastly, use the previous z=y/x relationship to replace the z portions and get your answer in terms of the original x and y variables.

My work is in the attachment. Comment woth questions or if something seems off.