1.\(f(x,y) \)= \( \) \( 3x^2 +4xy -y^2\)
\(f(tx, ty) \)= \( 3t^2+4t^2xy - 7t^2y^2\)
\(f(tx,ty) \)= \( t^2,(3x^2 +4xy -y^2) \)
= \( t^2f(x,y) \rightarrow \) homogen derajat 2
2. \(\ f(x,y) \)= \( \sqrt{x^2+y^2} \)
\(f(tx,ty) \)= \( tx+ \sqrt{t^2x^2+t^2y^2}\)
= \( tx\ +t \sqrt{x^2+y^2}\)
= \( t(x+ \sqrt{x^2+y^2}\)
= \( t.F(x,y)\rightarrow \) homogen derajat 1
3.\( (x+y)dx+xdy \)= \( 0\)
\(M(x,y) \)= \( x+y\)
\(M(tx, ty) \)= \( tx + ty \)= \( t(x+y) \)= \(t.M(x,y) \rightarrow \) homogen derajat 1
\(N(x)\)= \( x\)
\(n(tx)\)= \( tx\)
\(\)= \( t N (x) \rightarrow \) homogen derajat 1
\(N(x)\)= \( x \rightarrow \) PD Homogen
\(z=\frac{y}{z}\rightarrow y= xz\)
\(dy= zdx + xdz\)
\((x+xz)dx + x(zdx+xdz)=0\)
\((1+z)xdx+ zx dx+ x^{2}dz=0\)
\(x[1+z+z]dx + x^2dz=0\)
\(\frac{xdx}{x^2}+\frac{dz}{1+2z}=\int 0\)
\(=ln x + \frac{1}{2}ln (1+2z) =ln c\)
\(2lnc x +ln(1+2z)= 2 ln c\)
\(ln x^2 + ln(1+2z)=ln c^2\)
\(x^2.(1+2z)=c\)
\(x^2(1+2.\frac{x}{z}=c\)
\(x^2 +2xy=c\)
Contoh Soal Persamaan Diferensial Homogen
Selesai