Seriously wish you taught at my university. All of our higher class, calc3 and up, have really crappy instructors that care more about theory and proofs than actually showing us how to solve the problem. So a big thank you for helping those of us that need someone like you to break it down quick and simple in order to understand. If you have a patreon page or something let me know, you deserve some form of compensation for this. If it wasn't for you I would've failed a few different tests.
This is my first time watching one of your videos. I appreciate how you take your time with the problem and that you write very clearly (surprisingly hard to find). Using the two colors made it easier to follow. I learn a lot of my math from RU-vid and this was very helpful. Thank you.
Thank you! Helping me so much in preparing for the final in my DE class. Ever since I was in Calc 2 I have been watching these, and they are so helpful.
Without WolframAlpha, I know how to solve for v. Start by taking the exponential function of both sides, and relabeling the c: sec(v)+tan(v)=C₁*e^(-1/x). Change sec and tan to sin and cos and combine fractions: (1+sin(v))/cos(v)=C₁e^(1/x). Shift the sin and cos to cos and sin: (1+cos(v+π/2))/sin(v+π/2)=C₁e^(−1/x). Reciprocate: sin(v+π/2)/(1+cos(v+π/2))=C₂e^(1/x). Use the tangent half-angle identity: tan(v/2+π/4)=C₂e^(1/x). Take the arctan of both sides. v/2+π/4=arctan(C₂e^(1/x)). Solve for v: v=2arctan(C₂e^(1/x))−π/2. Substitute back in v=y/x²: y/x²=2arctan(C₂e^(1/x))−π/2. Solve for y: y=2x²arctan(C₂e^(1/x))−πx²/2. BAM! Solved for y.
Is there a special name for this sort of functions where you make a substitution to solve it? For example, homogenous 1st ODE you will substitute f(y/x)=f(v), v=y/x.
instead of using ln|sec(v)+tan(v)| for integral of sec(v) if we use artanh(sin(v)) (which is equivalent to ln|sec(v)+tan(v)), it makes it easy to solve for y i got y=x²arcsin(tanh(-1/x+c)) as final answer
“There’s no way to isolate the v”. Well even though int{sec z dz} = ln|sec z + tan z|, I think the more proper form to rewrite it with one input & no absolute value; +/-, is that the int{sec z dz} = arctanh(sin z). So with some “function sliding”, y = x^2 arcsin(tanh(c - 1/x)). Cool fun fact the derivative of arcsin(tanh z) = sech z
4:55 you could in fact solve for v. Let's say we have it in some final form like: sec(v) + tan(v) = r (r stands for whatever it is on the right-hand side) Then we do this: 1+sin(v) = cos (v) r 1 + (e^iv - e^-iv)/2i = (r/2) (e^iv + e^-iv) Now let e^iv be p. Then we have: 1 + (p + 1/p)/2i = (r/2) (p + 1/p) //multiply by 2 i p 2 i p + p^2 + 1 = r i p^2 + r i This is some polynomial in variable p, solve for p, make out a logarithm out of it and "see" the arctan function in it. Other approach is: rewrite sin(v) - r cos(v) as: sqrt(1+r^2) sin(x + arctan(r)) and you will finally obtain the same arctan formula. Of course, there would be some decisions like which root to take and add + 2*pi*integer somewhere when taking inverse functions, but, youknow, some people say: a differential equation is not complete unless you provide a sufficient set of initial and/or boundary conditions ;) so after you clarify initial condition, there should be no arbitrarity.
Could the answer be sec(y/x^2)+ tan(y/x^2) = Ce^(-1/x) ? Note : after removing the absolute value, i put plus or minus on the other side and a “plus or minus” constant is another constant.
instead of using ln|sec(v)+tan(v)| for integral of sec(v) if we use artanh(sin(v)) (which is equivalent to ln|sec(v)+tan(v)|), it makes it easy to solve for y i got y=x²arcsin(tanh(-1/x+c)) as final answer
instead of using ln|sec(v)+tan(v)| for integral of sec(v) if we use artanh(sin(v)) (which is equivalent to ln|sec(v)+tan(v)|), it makes it easy to solve for y i got y=x²arcsin(tanh(-1/x+c)) as final answer
