3.10-bo'lim : Yashirin farqlash
5. \(y'\) ni \(7{y^2} + \sin \left( {3x} \right) = 12 - {y^4}\) uchun yashirin farqlash orqali toping.
Maslahat: \(y\) haqiqatan ham \(y\left( x \right)\) ekanligini unutmang va shuning uchun biz \(y\) ishtirokidagi atamalarning hosilasini olishda zanjir qoidasidan foydalanishimiz kerak!
Yechim
Birinchidan, biz \(x\) ga nisbatan hamma narsaning hosilasini olishimiz kerak va biz \(y\) haqiqatan ham \(y\left( x \right)\) ekanligini esga olishimiz kerak, shuning uchun biz \(y\) ishtirokidagi atamalarning hosilasini olishda zanjir qoidasidan foydalanishimiz kerak.
\(x\) ga nisbatan farqlash shuni beradi,
\[14y\,y' + 3\cos \left( {3x} \right) = - 4{y^3}y'\]2-qadam
Va nihoyat, biz qilishimiz kerak bo'lgan yagona narsa buni \(y'\) uchun hal qilishdir.
\[\left( {14y + 4{y^3}} \right)y' = - 3\cos \left( {3x} \right)\hspace{0.5in} \Rightarrow \hspace{0.5in}\require{bbox} \bbox[2pt,border:1px solid black]{{y' = \frac{{ - 3\cos \left( {3x} \right)}}{{14y + 4{y^3}}}}}\]