Miscellaneous

Chapter 9 Class 12 Differential Equations
Serial order wise

### Transcript

Misc 8 Solve the differential equation π¦ π^(π₯/π¦) ππ₯= (π^(π₯/π¦)+π¦^2 )ππ¦ (π¦β 0)π¦ π^(π₯/π¦) ππ₯= (π^(π₯/π¦)+π¦^2 )ππ¦ ππ/ππ = (ππ^(π/π) + π^π)/(π^(π^(π/π) ) ) We can see that it is not homogeneous, so letβs try something else π¦π^(π₯/π¦) ππ₯/ππ¦=π₯π^(π₯/π¦)+π¦^2 π¦π^(π₯/π¦) ππ₯/ππ¦βπ₯π^(π₯/π¦)=π¦^2 π^(π₯/π¦) (π¦ ππ₯/ππ¦βπ₯)=π¦^2 π^(π/π) ((π ππ/ππ β π)/π^π )=π^π Let π^(π/π) = z Diff w.r.t. y. π^(π₯/π¦) π(π₯/π¦)/ππ¦ = ππ§/ππ¦ π^(π₯/π¦) ((π¦ ππ₯/ππ¦ β π₯ ππ¦/ππ¦)/π¦^2 )=ππ§/ππ¦ " " π^(π/π) ((π ππ/ππ β π)/π^π )=ππ/ππ " " From (1) ππ/ππ = 1 dz = dy Integrating on both sides β«1βππ§ = β«1βππ¦ z = y + c Putting π^(π₯/π¦) = z π^(π/π) = y + c

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#### Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.