In Problems 13 through 16, substitute y = e^rx into the given differential equat

Question

In Problems 13 through 16, substitute y = e^rx into the given differential equation to determine all values of the constant r for which y =e^rx is a solution of the equation. y" + y' - 2y = 0 In Problems 17 through 26, first verify that y(x) satisfies the given differential equation. Then determine a value of the constant C so that y(x) satisfies the given initial condition. Use a computer or graphing calculator (if desired) to sketch several typical solutions of the given differential equation, and high-light the one that satisfies the given initial condition. Y' + 3x^2 y = 0; y(x) = Ce^-x^3, y(0) = 7

Explanation / Answer

Solution ; the DE is y " + y ' - 2y = 0 . substitute y = e rx   , y ' = r erx  ,    y ' ' = r2 erx

: r2 erx + erx - 2erx  = 0 cancelling the common factor erx

; r2 + r - 2 = 0 = > r = - 2 and r = - 1

21. The DE is y ' + 3x2 y =0 . variable seperable method

y ' / y = - 3x2  . differentiate WRT x

log y = - x3 + log c where c is a constant of integration

y = c e - x^ 3 put x= 0 y =7 => c =7

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