Linear equations of first order. Solve the initial-value problem on the specified interval displaystyle{y}'-{3}{y}={e}^{{{2}{x}}}{o}{n}{left(-infty,+inftyright)}, text{with} {y}={0} text{when} {x}={0}.

Line 2021-03-08 Answered
Linear equations of first order.
Solve the initial-value problem on the specified interval y3y=e2xon(,+), with y=0 when x=0.
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Expert Answer

Nola Robson
Answered 2021-03-09 Author has 94 answers
From the given linear differential equation, P(x)=3andQ(x)=e2x.
Therefore, the integrating factor is given by
A(x)=0x3tdt=3
So that, the solution is given by
y=0e3x+e3x0xe3te2tdt
=e3x0xetdt
=e3x[et1]0x
=e3x(1ex)
=e3xe2x
We can verify the function is the solution of the initial-value periblem since it satisfies the differential equation and the initial condition.
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