Answer to Question #197375 in Differential Equations for fhumulani raphunga

Question #197375

(D^2-2D+3)y=X^2-1


1
Expert's answer
2021-05-24T15:20:46-0400

Let us solve the differential equation (D22D+3)y=x21(D^2-2D+3)y=x^2-1. For this let us solve the characteristic equation of the homogeneous equation (D22D+3)y=0(D^2-2D+3)y=0:


k22k+3=0k^2-2k+3=0


(k1)2+2=0(k-1)^2+2=0


(k1)2=2(k-1)^2=-2


k1=1+i2k_1=1+i\sqrt{2} and k2=1i2.k_2=1-i\sqrt{2}.


It follows that the general solution of the equation (D22D+3)y=x21(D^2-2D+3)y=x^2-1 is of the form y=ex(C1cos(2x)+C2sin(2x))+yp,y=e^x(C_1\cos(\sqrt{2}x)+ C_2\sin(\sqrt{2}x))+y_p, where yp=Ax2+Bx+C.y_p=Ax^2+Bx+C. It follows that yp=2Ax+By_p'=2Ax+B and yp=2A.y_p''=2A. Therefore, 2A2(2Ax+B)+3(Ax2+Bx+C)=x21.2A-2(2Ax+B)+3(Ax^2+Bx+C)=x^2-1. We conclude that 3Ax2+(3B4A)x+(2A2B+3C)=x21.3Ax^2+(3B-4A)x+(2A-2B+3C)=x^2-1. Consequently, 3A=1, 3B4A=0, 2A2B+3C=1.3A=1,\ 3B-4A=0,\ 2A-2B+3C=-1. It follows that A=13, B=49, C=13(123+89)=727.A=\frac{1}{3},\ B=\frac{4}{9},\ C=\frac{1}{3}(-1-\frac{2}{3}+\frac{8}{9})=-\frac{7}{27}.


We conclude that the general solution of the differential equation (D22D+3)y=x21(D^2-2D+3)y=x^2-1 is


y=ex(C1cos(2x)+C2sin(2x))+13x2+49x727.y=e^x(C_1\cos(\sqrt{2}x)+ C_2\sin(\sqrt{2}x))+\frac{1}{3}x^2+\frac{4}{9}x-\frac{7}{27}.



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