The differential equation \frac{d y}{d x}+4 y=5 is valid in the domain 0 \leq x \leq 1 with y(0)=2.25. The solution of the differential equation is

Correct Answer is : y=e−4x+1.25

GATE Mechanical Engineering (ME) 2019 Shift 2 Solved Paper

Question : 34 of 65

Marks: +1, -0

The differential equation

\;{d y}/{d x}+4 y=5

is valid in the domain

0 ≤ x ≤ 1

with

y(0)=2.25

. The solution of the differential equation is

Solution:

The given equation is.

\;{d y}/{d x}+4 y=5

If Integration factor

=e^{∫ 4 d x}=e^{4 x}

⇒ y × I F=∫ 5 × I F d x+C

⇒ y × e^{4 x}=5 × \;{e^{4 x}}/{4}+C

⇒ y × e^{4 x}=5 × \;{e^{4 x}}/{4}+C . . . . .(1)

Now for at

x=0

⇒ 2.25=\;{5}/{4}+C

⇒ C=1

\Rightarrow

Using the value of

C

in equ

(1)

⇒ y × \;{e^{4 x}}/{e^{4 x}}=\;{5}/{e^{4 x}} × \;{e^{4 x}}/{4}+\;{c}/{e^{4 x}}

⇒ y=1.25+e^{-4 x}

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