Explanation Given: d y d t = y + t 2 and y ( 0 ) = 4 Formula used: ∫ x n d x = 1 n + 1 x n + 1 ……………. (1) ∫ ( u ( x ) . v ( x ) ) d x = u ( x ) ∫ v ( x ) d x − ∫ u ' ( x ) ∫ v ( x ) d x …………….(2) ∫ e x d x = e x + c …………… (3) Calculation: Consider the given equation d y d t = y + t 2 According to the rule of linear differential equations, every linear differential equation can be represented in the form of: d y d x + P . y = Q where P and Q are functions of x and integrating factor I . F = e ∫ P d x And solution is e ∫ P d x . y = ∫ Q . e ∫ P d x d x Converting given equation into standard form d y d t − y = + t 2 P = − 1 and Q = t 2 I . F = e ∫ − 1 d t ⇒ I . F = e − t ………………(using equation 3) Now general solution is equal to e − t . y = ∫ t 2 ...

4th Edition

ISBN: 9781319323394

Author: Rogawski

Publisher: MAC HIGHER

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Differential Equations

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The general solution of d y d t = y + t 2

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