S o l u t i o n Solution S o l u t i o n
( 2 D 3 + 3 D 2 D ′ + 4 D D ′ 2 + 5 D ′ 3 ) z = x 2 + y (2D3+3D2D'+4DD'2+5D'3)z=x2+y ( 2 D 3 + 3 D 2 D ′ + 4 D D ′ 2 + 5 D ′ 3 ) z = x 2 + y
The PDE is a homogeneous linear differential equation,thus auxilary equation is;
2 m 3 + 3 m 2 + 4 m + 5 = 0 2m3+3m2+4m+5=0 2 m 3 + 3 m 2 + 4 m + 5 = 0 ∴ ⟹ m 1 = 1 2 ( − 1 + ( − 63 + 2 1086 ) 1 / 3 3 2 / 3 − 5 ( 3 ( − 63 + 2 1086 ) ) 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}})\\ ∴ ⟹ m 1 = 2 1 ( − 1 + 3 2/3 ( − 63 + 2 1086 ) 1/3 − ( 3 ( − 63 + 2 1086 ) ) 1/3 5 )
m 2 = − 1 2 − ( 1 + ( 1 + i 3 ) ( − 63 + 2 1086 ) 1 / 3 4 ∗ 3 2 / 3 + 5 ( 1 − i 3 ) 4 ( 3 ( − 63 + 2 1086 ) ) 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}})\\ m 2 = − 2 1 − ( 1 + 4 ∗ 3 2/3 ( 1 + i 3 ) ( − 63 + 2 1086 ) 1/3 + 4 ( 3 ( − 63 + 2 1086 ) ) 1/3 5 ( 1 − i 3 ) )
m 3 = − 1 2 − ( 1 + ( 1 − i 3 ) ( − 63 + 2 1086 ) 1 / 3 4 ∗ 3 2 / 3 + 5 ( 1 + i 3 ) 4 ( 3 ( − 63 + 2 1086 ) ) 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}})\\ m 3 = − 2 1 − ( 1 + 4 ∗ 3 2/3 ( 1 − i 3 ) ( − 63 + 2 1086 ) 1/3 + 4 ( 3 ( − 63 + 2 1086 ) ) 1/3 5 ( 1 + i 3 ) )
H e n c e , Hence, He n ce ,
C . F = f 1 ( y + m 1 x ) + f 2 ( y + m 2 x ) + f 3 ( y + m 3 x ) C.F=f_1(y+m_1x)+f_2(y+m_2x)+f_3(y+m_3x) C . F = f 1 ( y + m 1 x ) + f 2 ( y + m 2 x ) + f 3 ( y + m 3 x )
N o w ; Now; N o w ;
( P . I ) 1 = x 2 2 D 3 + 3 D 2 D ′ + 4 D D ′ 2 + 5 D ′ 3 (P.I)_1=\frac{x^2}{2D^3+3D^2D′+4DD′^2+5D′^3} ( P . I ) 1 = 2 D 3 + 3 D 2 D ′ + 4 DD ′ 2 + 5 D ′ 3 x 2
⟹ 1 2 D 3 ( 1 + ( 3 D ′ 2 D + 2 ( D ′ D ) 2 + 5 2 ( D ′ D ) 2 ) ) − 1 ( x 2 ) = 1 2 D 3 ( x 2 ) = x 5 120 \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} ⟹ 2 D 3 1 ( 1 + ( 2 D 3 D ′ + 2 ( D D ′ ) 2 + 2 5 ( D D ′ ) 2 ) ) − 1 ( x 2 ) = 2 D 3 1 ( x 2 ) = 120 x 5
F o r ( P . I ) 2 ; For (P.I)_2; F or ( P . I ) 2 ;
( P . I ) 2 = y 2 D 3 + 3 D 2 D ′ + 4 D D ′ 2 + 5 D ′ 3 (P.I)_2=\frac{y}{2D^3+3D^2D′+4DD′^2+5D′^3} ( P . I ) 2 = 2 D 3 + 3 D 2 D ′ + 4 DD ′ 2 + 5 D ′ 3 y
⟹ 1 5 D ′ 3 ( 1 + ( 4 D 5 D ′ + 3 5 ( D D ′ ) 2 + 2 5 ( D D ′ ) 2 ) ) − 1 ( y ) = 1 5 D ′ 3 ( y ) = y 4 120 \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} ⟹ 5 D ′3 1 ( 1 + ( 5 D ′ 4 D + 5 3 ( D ′ D ) 2 + 5 2 ( D ′ D ) 2 ) ) − 1 ( y ) = 5 D ′3 1 ( y ) = 120 y 4
H e n c e , Hence, He n ce ,
C . F + ( P . I ) 1 + ( P . I ) 2 C.F+(P.I)_1+(P.I)_2 C . F + ( P . I ) 1 + ( P . I ) 2 ⟹ ∑ i = 1 3 f i ( y + m i x ) + 1 120 ( x 5 + y 4 ) \implies ∑_{i=1}^{3}f_i(y+m_ix)+\frac{1}{120}(x^5+y^4) ⟹ ∑ i = 1 3 f i ( y + m i x ) + 120 1 ( x 5 + y 4 )
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