Solving Linear Differential Equations of First Order
Let y = y x b...
Question
Let y=y(x) be the solution of the differential equation dydx+2y=f(x), where f(x)={1,x∈[0,1]0,otherwise If y(0)=0, then y(32) is
A
e2+12e4
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B
12e
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C
e2−1e3
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D
e2−12e3
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Solution
The correct option is De2−12e3 dydx+2y=f(x) Integrating factor, I.F.=e∫2dx=e2x ⇒y⋅e2x=∫e2xf(x)dx+C⇒y=e−2x∫e2xf(x)dx+Ce−2x As y(0)=0 ⇒y(x)=e−2xx∫0e2xf(x)dx⇒y(32)=e−33/2∫0e2xf(x)dx⇒y(32)=e−31∫0e2x⋅1dx+e−33/2∫1e2x⋅0dx⇒y(32)=[e2x]102e3=e2−12e3