How to solve the differential-integral equation?

Question

$phi'(t)=1/2int_0^t(t-sigma)^2phi(sigma)dsigma=-t$ $phi(0)=1$

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Answer 1 · 2018-06-28T20:43:39

$phi(t) = 1 - t^2/2$

${(phi'(t)=-t qquad triangle),(phi'(t) = 1/2 int_0^t(t-sigma)^2phi(sigma)dsigma qquad square):}$

$triangle implies phi(t) = - t^2 /2 + C$

$phi(0) = 1 implies phi(t) = 1 - t^2/2$