Define Initial Value Problem. Solve that value problem of y2+5y=1, y(0) = 2

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Suresh Chand · Accepted Answer

Definition of Initial Value Problem: A differential equation together with initial condition(s) is called the initial value problem. For example, (frac{d^2y}{d^2x} = 2frac{dy}{dx} + y = 0, y(0) = -1, y^{'}(0) = 1) Here (y(0) = -1, y^{'}(0) = 1) is an initial conditions   Problem Part: Solution: Given that y' + 5y = 1          ...... (i) y(0) = 2             .......(ii) Since (i) is linear differential equation of first order. Comparing (i) with y' + Py = Q then P = 5, Q = 1 Then (int P dx = 5 int dx = 5x) Here, the integrating factor (I.F) of (i) is (I.F. = e^{int Pdx} = e^{5x}) Now, Multiplying (i) by I.F. and then integrating we get, (y times I.F. = int(Q times I.F.)dx + C) (i.e enspace ye^{5x} = int (1) (e^{5x}) dx + C = frac{e^{5x}}{5} + C)           .....(iii) By (ii), we have y(0) = 2, then (iii) becomes, (ye^{5x} = frac{e^{5x}}{5} + frac{9}{5}) (Rightarrow y = frac{1}{5} (1 + 9e^{-5x})) This is the solution of (i) that stratifies (ii).

Define Initial Value Problem. Solve that value problem of \(y^2 + 5y = 1\), y(0) = 2

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Define Initial Value Problem. Solve that value problem of \(y^2 + 5y = 1\), y(0) = 2

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