Solve this exact differential equation

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  • #1
chwala
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Homework Statement:
Solve the exact differential equation
Relevant Equations:
exact equations
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1614344793714.png


now my approach is different, i just want to check that my understanding on this is correct.

see my working below;
 
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Answers and Replies

  • #2
chwala
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##2xy-9x^2+(2y+x^2+1)\frac {dy}{dx}=0##
##2xy-9x^2dx+(2y+x^2+1)dy=0##
Let ##M(x,y)=2xy-9x^2##
##N(x,y)=2y+x^2+1## Since ##\frac {∂M}{∂y}=2x=\frac {∂N}{∂x}=## then the differential equation is exact.
Therefore, ##\int Mdx## = ##x^2y-3x^3+F(y)##........1
and ##\int Ndy## = ##y^2+x^2y+y+c ##......2
therefore, ##F(y)= y^2+y+c##......3

therefore, we shall have (from 1 and 3), ## x^2y-3x^3+y^2+y=c##
i understand it this way better, i just want to know if this is also correct.
 
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  • #3
PeroK
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Looks fine to me.
 
  • #4
chwala
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That's how I understand it better from my undergraduate studies...thanks
 
  • #5
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i just want to know if this is also correct.
Once you have your solution, it's good practice to check by finding the total derivative of your expression, which you should be able to manipulate back into the form the equation was given in.
 
  • #6
chwala
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The solution is correct, its a textbook question...my interest was on the approach or rather my way of working the problem to realize the solution.
Thanks Mark for your input. Yeah I will use total derivatives to check the solution...
 
  • #7
chwala
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Once you have your solution, it's good practice to check by finding the total derivative of your expression, which you should be able to manipulate back into the form the equation was given in.

just to follow your guidance, on checking...
let ##u=x^2y-3x^3+y^2+y##
##f_{x}=2xy-9x^2##
##f_{y}=x^2+2y+1##
therefore,
##du=f_{x} dx+f_{y} dy##
##du=(2xy-9x^2)dx+(x^2+2y+1)dy##
bingo!
 

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