Answer:
1) [tex]y = -\frac{32}{5}x^{-3}-\frac{8}{5}x^7[/tex]
2) [tex]y=\left(2-\frac{7\sqrt{13}+26}{26}\right)x^{3+\sqrt{13}}+\frac{\left(7\sqrt{13}+26\right)}{26}x^{3-\sqrt{13}}[/tex]
Step-by-step explanation:
For both of these questions, we will use the Reduction of Orders rule of the Second Order Differential Equations.
1) [tex]x^2y''-3xy'-21y=0[/tex] where [tex]y(1)=-8[/tex] and [tex]y'(1)=8[/tex]
Let [tex]y=vy_1[/tex] where v is a function of x. Then,
[tex]y_2'=v'y_1+vy_1'\\y_2''=v''y_1'+v'y_1'+v'y_1'+vy_1''=v''y_1'+2v'y_1'+vy_1''[/tex]
Insert them into the equation,
[tex]v''y_1'+2v'y_1'+vy_1'-\frac{3}{x}(v'y_1+vy_1')-\frac{21}{x^2}(vy_1)=0\\y_1v''+(2y_1'-\frac{3}{x}y_1)v'+(y_1''-\frac{3}{x}y_1'-\frac{21}{x^2}y_1)v=0[/tex]
Choose [tex]y_1 = x^7[/tex] which eliminates [tex](y_1''-\frac{3}{x}y_1'-\frac{21}{x^2}y_1)[/tex]
Hence,
[tex]x^7v''+(2*7x^6-\frac{3}{x}x^7)v'=0\\x^7v''+11x^6v'=0[/tex]
Let [tex]w = v'[/tex], then [tex]w'=v''[/tex]
Hence,
[tex]x^7w'=-11x^6w[/tex] ⇒ [tex]\frac{dw}{dx}=\frac{-11}{x}w[/tex] ⇒ [tex]\frac{dw}{w}=\frac{-11dx}{x}[/tex]
Now let's take the integral of both sides and then use logarithm rule,
[tex]ln(w)=-11ln(x) + C_0\\e^{ln(w)}=e^{-11ln(x) + C_0}\\w=C_1x^{-11}=v'\\v=\frac{C_1x^{-10}}{-10}+C_2\\v=Cx^{-10}+C_2[/tex]
Thus,
[tex]y = vy_1=(Cx^{-10}+C_2)x^7=Cx^{-3}+C_2x^7[/tex]
Use initial conditions now,
[tex]C+C_2=-8\\-3C+7C_2=8[/tex]
Then [tex]C = -\frac{32}{5}[/tex] and [tex]C_2 = -\frac{8}{5}[/tex]
Finally,
[tex]y = -\frac{32}{5}x^{-3}-\frac{8}{5}x^7[/tex] is general solution.
2) [tex]x^2y''-5xy'-4y=0[/tex] where [tex]y(1)=2[/tex] and [tex]y'(1)=-1[/tex]
[tex]y''-\frac{5}{x}y'-\frac{4}{x^2}y=0[/tex]
Let's use [tex]y=x^r[/tex]
[tex](x^r)''-\frac{5}{x}(x^r)'-\frac{4}{x^2}x^r=0\\x^{r-2}(r^2-6r-4)=0[/tex]
So roots are [tex]r_1=3+\sqrt{13},\:r_2=3-\sqrt{13}[/tex]
[tex]y = c_1x^{3+\sqrt{13}}+c_2x^{3-\sqrt{13}}[/tex]
By applying initial conditions, constants are found as follows,
[tex]c_1=\left(2-\frac{7\sqrt{13}+26}{26}\right),\:c_2=\frac{\left(7\sqrt{13}+26\right)}{26}[/tex]
Hence,
[tex]y=\left(2-\frac{7\sqrt{13}+26}{26}\right)x^{3+\sqrt{13}}+\frac{\left(7\sqrt{13}+26\right)}{26}x^{3-\sqrt{13}}[/tex] is general solution
