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# clase4handouts - Leibniz Integral rule-Necessary for...

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Leibniz Integral rule-Necessary for homework The Leibniz integral rule gives a formula for differentiation of a deﬁnite integral whose limits are functions of the differential variable, z Z b ( z ) a ( z ) f ( x , z ) dx = Z b ( z ) a ( z ) f z dx + f ( b ( z ) , z ) db dz - f ( a ( z ) , z ) da dz . Therefore, t Z t 0 e - 2 ( s 2 - t 2 ) ds = Z t 0 4 te - 2 ( s 2 - t 2 ) ds + e - 2 ( t 2 - t 2 ) dt dt - e - 2 ( 0 2 - t 2 ) da dt . = Z t 0 4 te - 2 ( s 2 - t 2 ) ds + 1 . M.I. Bueno Differential Equations and Linear Algebra

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Quantitative analysis We use the method of separation of variables to compute explicit or implicit solutions of ﬁrst-order differential equations of the form y 0 = f ( t ) g ( y ) or y 0 = f ( t ) / h ( y ) . This kind of equations are called separable differential equations . Constant solutions can be found by solving g ( y ) = 0. M.I. Bueno Differential Equations and Linear Algebra
Example Find the general solution of e y y 0 - ( t + t 3 ) = 0. This equation can be written in the form e y y 0 = t + t 3 . Notice that this differential equation is separable and has no equilibrium solutions. Multiplying both sides by dt , e y dy = ( t + t 3 ) dt , Integrating both sides Z e y dy = Z ( t + t 3 ) dt , e y = t 2 2 + t 4 4 + C . Applying logarithms to both sides, y = ln ( t 2 2 + t 4 4 + C ) . Explicit solution. M.I. Bueno Differential Equations and Linear Algebra

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Example Find the general solution of y 0 = t 2 / ( 1 - y 2 ) . 1
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clase4handouts - Leibniz Integral rule-Necessary for...

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