Higher order math

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The document introduces a presentation about homogeneous differential equations. It provides the name and ID of the presenter, then gives an example of a homogeneous differential equation dy/dx = y/(x+xy). The solution steps are shown, applying the property that a homogeneous function of degree zero is unchanged when its variables are multiplied by a constant. Through separation of variables and integration, the solution to the differential equation is obtained as -y/x = ln(y).

$WELCOME TO MY PRESENTATION$

$MY PRESENTATION TOPIC IS HOMOGENEOUS NAME: PIAS AHMMED ID:151-15-5198$

$SOLVE THE FOLLOWING HOMOGENEOUS DE 𝑑𝑦 𝑑𝑥 = 𝑦 𝑥+ 𝑥𝑦 Sol: The given DE is 𝑑𝑦 𝑑𝑥 = 𝑦 𝑥+ 𝑥𝑦 Let, f(x,y)= 𝑦 𝑥+ 𝑥𝑦 f(tx,ty)=𝑡0( 𝑦 𝑥+ 𝑥𝑦 ) =𝑡0 𝑓(𝑥, 𝑦) f(x , y) is a homogenous function of Zero degree. So the given DE is also a homogenous DE.$

$Let us consider y= vx Or, 𝑑𝑦 𝑑𝑥 = v+x 𝑑𝑦 𝑑𝑥 Or, 𝑣𝑥 𝑥+ 𝑥2 𝑣 = v+x 𝑑𝑣 𝑑𝑥 Or, 𝑣𝑥 𝑥(1+ 𝑣) =v+ x 𝑑𝑣 𝑑𝑥 Or, 𝑣 1+ 𝑣 = 𝑣 + 𝑥 𝑑𝑣 𝑑𝑥 Or, 𝑣 + 𝑥 𝑑𝑣 𝑑𝑥 = 𝑣 1+ 𝑣 Or,𝑥 𝑑𝑣 𝑑𝑥 = 𝑣 1+ 𝑣 − 𝑣 Or, 𝑥 𝑑𝑣 𝑑𝑥 = 𝑣−𝑣−𝑣. 𝑣 1+ 𝑣 Or, 𝑥 𝑑𝑣 𝑑𝑥 = −𝑣 𝑣 1+ 𝑣$

$Separating the variables and integrating Or, ( 1+ 𝑣 −𝑣. 𝑣 )𝑑𝑣= 1 𝑥 𝑑𝑥 Or,( 1 −𝑣 𝑣 − 𝑣 𝑣 𝑣 )𝑑𝑣= 1 𝑥 𝑑𝑥 Or, 1 −𝑣 𝑣 𝑑𝑣 − 1 𝑣 𝑑𝑣= 1 𝑥 𝑑𝑥 Or, 1𝑑𝑣 − 1 𝑣 𝑑𝑣= 1 𝑥 𝑑𝑥 Or, -v-lnv=lnx Or, -v=lnvx Or,- 𝑦 𝑥 =lny$

$THANK YOU ALL….!!!$

Higher order math

1. WELCOME TO MY PRESENTATION

2. MY PRESENTATION TOPIC IS HOMOGENEOUS NAME: PIAS AHMMED ID:151-15-5198

3. SOLVE THE FOLLOWING HOMOGENEOUS DE 𝑑𝑦 𝑑𝑥 = 𝑦 𝑥+ 𝑥𝑦 Sol: The given DE is 𝑑𝑦 𝑑𝑥 = 𝑦 𝑥+ 𝑥𝑦 Let, f(x,y)= 𝑦 𝑥+ 𝑥𝑦 f(tx,ty)=𝑡0( 𝑦 𝑥+ 𝑥𝑦 ) =𝑡0 𝑓(𝑥, 𝑦) f(x , y) is a homogenous function of Zero degree. So the given DE is also a homogenous DE.

4. Let us consider y= vx Or, 𝑑𝑦 𝑑𝑥 = v+x 𝑑𝑦 𝑑𝑥 Or, 𝑣𝑥 𝑥+ 𝑥2 𝑣 = v+x 𝑑𝑣 𝑑𝑥 Or, 𝑣𝑥 𝑥(1+ 𝑣) =v+ x 𝑑𝑣 𝑑𝑥 Or, 𝑣 1+ 𝑣 = 𝑣 + 𝑥 𝑑𝑣 𝑑𝑥 Or, 𝑣 + 𝑥 𝑑𝑣 𝑑𝑥 = 𝑣 1+ 𝑣 Or,𝑥 𝑑𝑣 𝑑𝑥 = 𝑣 1+ 𝑣 − 𝑣 Or, 𝑥 𝑑𝑣 𝑑𝑥 = 𝑣−𝑣−𝑣. 𝑣 1+ 𝑣 Or, 𝑥 𝑑𝑣 𝑑𝑥 = −𝑣 𝑣 1+ 𝑣

5. Separating the variables and integrating Or, ( 1+ 𝑣 −𝑣. 𝑣 )𝑑𝑣= 1 𝑥 𝑑𝑥 Or,( 1 −𝑣 𝑣 − 𝑣 𝑣 𝑣 )𝑑𝑣= 1 𝑥 𝑑𝑥 Or, 1 −𝑣 𝑣 𝑑𝑣 − 1 𝑣 𝑑𝑣= 1 𝑥 𝑑𝑥 Or, 1𝑑𝑣 − 1 𝑣 𝑑𝑣= 1 𝑥 𝑑𝑥 Or, -v-lnv=lnx Or, -v=lnvx Or,- 𝑦 𝑥 =lny

6. THANK YOU ALL….!!!

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Higher order math

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Higher order math