Question #126198
Solve (2D^3+3D^2D'+4DD'^2+5D'^3)z=x^2+y
1
Expert's answer
2020-07-14T17:54:39-0400

SolutionSolution


(2D3+3D2D+4DD2+5D3)z=x2+y(2D3+3D2D'+4DD'2+5D'3)z=x2+y


The PDE is a homogeneous linear differential equation,thus auxilary equation is;



2m3+3m2+4m+5=02m3+3m2+4m+5=0

      m1=12(1+(63+21086)1/332/35(3(63+21086))1/3)\therefore   \implies m_1=\frac{1}{2} (-1 + \frac{(-63 + 2\sqrt{1086})^{1/3}}{3^{2/3}} - \frac{5}{(3 (-63 + 2\sqrt{1086}))^{1/3}})\\

m2=12(1+(1+i3)(63+21086)1/3432/3+5(1i3)4(3(63+21086))1/3)m_2=-\frac{1}{2}- (1 + \frac{(1+i\sqrt{3})(-63 + 2\sqrt{1086})^{1/3}}{4*3^{2/3}} + \frac{5(1-i\sqrt{3})}{4(3 (-63 + 2\sqrt{1086}))^{1/3}})\\

m3=12(1+(1i3)(63+21086)1/3432/3+5(1+i3)4(3(63+21086))1/3)m_3=-\frac{1}{2}- (1 + \frac{(1-i\sqrt{3})(-63 + 2\sqrt{1086})^{1/3}}{4*3^{2/3}} + \frac{5(1+i\sqrt{3})}{4(3 (-63 + 2\sqrt{1086}))^{1/3}})\\

Hence,Hence,


C.F=f1(y+m1x)+f2(y+m2x)+f3(y+m3x)C.F=f_1​(y+m_1​x)+f_2​(y+m_2​x)+f_3​(y+m_3​x)

Now;Now;

(P.I)1=x22D3+3D2D+4DD2+5D3(P.I)_1​=\frac{x^2}{2D^3+3D^2D′+4DD′^2+5D′^3​}


    12D3(1+(3D2D+2(D′​D)2+52(DD)2))1(x2)=12D3(x2)=x5120\implies \frac{1}{2D^3​}(1+(\frac{3D'}{2D}​+2(\frac{D′​}{D})^2+\frac{5}{2}​(\frac{D'}{D})^2))^{−1}(x^2)=\frac{1}{2D^3​}(x^2)=\frac{x^5​}{120}


For(P.I)2;For (P.I)_2;

(P.I)2=y2D3+3D2D+4DD2+5D3(P.I)_2​=\frac{y}{2D^3+3D^2D′+4DD′^2+5D′^3​}


    15D3(1+(4D5D+35(DD)2+25(DD)2))1(y)=15D3(y)=y4120\implies \frac{1}{5D'^3​}(1+(\frac{4D}{5D'}​+\frac{3}{5}(\frac{D​}{D'})^2+\frac{2}{5}​(\frac{D}{D'})^2))^{−1}(y)=\frac{1}{5D'^3​}(y)=\frac{y^4}{120}


Hence,Hence,


C.F+(P.I)1+(P.I)2C.F+(P.I)_1​+(P.I)_2​

    i=13fi(y+mix)+1120(x5+y4)\implies ∑_{i=1}^{3}​f_i​(y+m_i​x)+\frac{1}{120}​(x^5+y^4)






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