Given that [tex]y(0)=0[/tex] and [tex]y'(0)=7[/tex], we have
[tex]\begin{cases}y(x)=C_1e^x+C_2e^{-x}\\y'(x)=C_1e^x-C_2e^{-x}\end{cases}[/tex]
[tex]\implies\begin{cases}0=C_1+C_2\\7=C_1-C_2\end{cases}[/tex]
[tex]\implies C_1=\dfrac72,C_2=-\dfrac72[/tex]
so that the particular solution is
[tex]\boxed{y(x)=\dfrac72e^x-\dfrac72e^{-x}}[/tex]
Equations which involves functions and its derivative is known as differential equation. these equations are highly used to model behaviour of complex system.
y(x) = [tex]C_{1} *e^{x}[/tex] + [tex]C_{2} *e^{-x}[/tex]
y(0) = [tex]C_{1} + C_{2}[/tex]
0 = [tex]C_{1} + C_{2}[/tex] --------------------------------------------------(1)
y'(x) = [tex]C_{1} *e^{x}[/tex] - [tex]C_{2} *e^{-x}[/tex]
y'(0) = [tex]C_{1} - C_{2}[/tex]
7 = [tex]C_{1} - C_{2}[/tex] -------------------------------------------------(2)
From Equation 1 and 2
[tex]C_{1} + C_{2}[/tex] = 0
[tex]C_{1} - C_{2}[/tex] = 7
2[tex]C_{1}[/tex] = 7
[tex]C_{1}[/tex] = [tex]\frac{7}{2}[/tex]
and [tex]C_{2}[/tex] = - [tex]\frac{7}{2}[/tex]
Thus, y = [tex]\frac{7}{2}[/tex] [tex]e^{x}[/tex] - [tex]\frac{7}{2}[/tex] [tex]e^{x}[/tex]
To learn more about differential equations visit:
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