Answer:
The solution of given differential equation [tex]y = 3 e^{\frac{1}{2}x }[/tex]
Step-by-step explanation:
Step1:-
Given differential equation
[tex]\frac{dy}{dx} = \frac{1}{2}y[/tex]
The differential operator form [tex](D - \frac{1}{2})y =0[/tex]
The auxiliary equation is(A.E) f(m) = m - 1/2 =0
m = 1/2
The complementary solution is [tex]y(x) = c_{1} e^{a_{1} x} + c_{2} e^{a_{2} x}[/tex]
The complementary solution is [tex]y(x) = c_{1} e^{\frac{1}{2}x }[/tex] .......(1)
Step 2:-
Given conditions are x =0 and y(0) =3
From (1) we get
[tex]y(0) = c_{1} e^{\frac{1}{2}0 }[/tex]
[tex]3 = c_{1}[/tex]
now the solution of the given differential equation
substitute [tex]3 = c_{1}[/tex] in equation(1) , we get
[tex]y(x) = 3 e^{\frac{1}{2}x }[/tex]
Final answer :-
The solution of given differential equation [tex]y = 3 e^{\frac{1}{2}x }[/tex]
