Respuesta :
Answer:
[tex]x\left(\frac{7}{4}\right)=2.875[/tex].
Step-by-step explanation:
The Euler's method states that [tex]y_{n+1}=y_n+h \cdot f \left(x_n, y_n \right)[/tex], where [tex]x_{n+1}=x_n + h[/tex].
We want to find [tex]x(1.75)=x\left(\frac{7}{4}\right)[/tex] for [tex]\frac{dx}{dt}=2x[/tex], when [tex]t_0=1[/tex], [tex]x_0=1[/tex], and [tex]h=0.25[/tex] using the Euler's method.
Step 1.
[tex]t_{1}=t_{0}+h=1+\frac{1}{4}=\frac{5}{4}[/tex]
[tex]x\left(t_{1}\right)=x\left( \frac{5}{4} \right)=x_{1}=x_{0}+h \cdot f \left(t_{0}, x_{0} \right)=1+h \cdot f \left(1, 1 \right)[/tex]
[tex]x\left(t_{1}\right)=1 + \frac{1}{4} \cdot \left(2.0 \right)=1.5[/tex]
Step 2.
[tex]t_{2}=t_{1}+h=\frac{5}{4}+\frac{1}{4}=\frac{3}{2}[/tex]
[tex]x\left(t_{2}\right)=x\left( \frac{3}{2} \right)=x_{2}=x_{1}+h \cdot f \left(t_{1}, x_{1} \right)=1.5+h \cdot f \left(\frac{5}{4}, 1.5 \right)[/tex]
[tex]x\left(x_{2}\right)=1.5 + \frac{1}{4} \cdot \left(2.5 \right)=2.125[/tex]
Step 3.
[tex]t_{3}=t_{2}+h=\frac{3}{2}+\frac{1}{4}=\frac{7}{4}[/tex]
[tex]x\left(t_{3}\right)=x\left( \frac{7}{4} \right)=x_{3}=x_{2}+h \cdot f \left(t_{2}, x_{2} \right)=2.125+h \cdot f \left(\frac{3}{2}, 2.125 \right)[/tex]
[tex]x\left(t_{3}\right)=2.125 + \frac{1}{4} \cdot \left(3.0 \right)=2.875[/tex]
