An explicit forward Euler method is used to numerically integrate the differential equation \frac{d y}{d t}=y Using a time step of 0.1. With the initial condition y(0)=1, the value of y(1) computed by this method is (correct to two decimal places).

GATE Mechanical Engineering (ME) 2018 Shift 1 Solved Paper

Question : 56 of 65

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An explicit forward Euler method is used to numerically integrate the differential equation

\;{d y}/{d t}=y

Using a time step of

0.1

. With the initial condition

y(0)=1

, the value of

y(1)

computed by this method is (correct to two decimal places).

Solution:

y_{1} \;\; =y_{0}+h_{f}(t_{0}, y_{0})

\;\; =y_{0}+h y_{0}

\;\; =1+0.1(1)

y_{1} \;\; =1.1

y_{2} \;\; =y_{1}+h_{f}(t_{1}, y_{1})

\;\; =y_{1}+h ⋅ y_{1}

\;\; =1.1+0.1(1.1)

y_{2} \;\; =1.21

y_{3} \;\; =y_{2}+h_{f}(t_{2}, y_{2})

\;\; =y_{2}+h ⋅ y_{2}

\;\; =1.21+0.1 × 1.21

y_{3} \;\; =1.331

y_{4} \;\; =y_{3}+h . f(t_{3}, y_{3})

\;\; =y_{3}+h ⋅ y_{3}

\;\; =1.331+0.1 × 1.331

y_{4} \;\; =1.4641

y_{5} \;\; =y_{4}+h ⋅ f(t_{4}, y_{4})

\;\; =y_{4}+h ⋅ y_{4}

\;\; =1.4641+0.1 ×(1.4641)

y_{5} \;\; =1.61051

y_{6} \;\; =y_{5}+\;\text " h.f "\;(t_{5}, y_{5})

\;\; =y_{5}+\;\text " h. "\; y_{5}

\;\; =1.61051+0.1 × 1.61051

y_{6} \;\; =1.771561

y_{7} \;\; =y_{6}+h . f(t_{6}, y_{6})

\;\; =y_{6}+h × y_{6}

\;\; =1.771561+0.1 × 1.771561=1.9487

y_{8} \;\; =y_{7}+h . f(t_{7}, y_{7})

\;\; =y_{7}+h ⋅ y_{7}

\;\; =1.9487+0.1 ×(1.9487)

y_{8} \;\; =2.14357

y_{9} \;\; =y_{8}+h ⋅ f(t_{8}, y_{8})

\;\; =y_{8}+h ⋅ y_{8}

\;\; =2.14357+0.1 × 2.14357

y_{9} \;\; =2.3579

y_{10}=y_{9}+h ⋅ f(t_{9}, y_{9})

\;\; =y_{9}+h . y_{9}

\;\; =2.3579+0.1 ×(2.3579)

y_{10}=2.5937

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