Differential equationsShow that y = 2/3e^(x) + e^(-2x)
3 months ago
Show that y = 2/3e^(x) + e^(-2x) is a solution of the differential equation y’ + 2y = 2e^(x)
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Best Answer
bushticketscolorado Staff answered 3 months ago
Step 1 y=(2)/(3)e^(x)+e^(-2x) Differentiate with respect to x y'=(2)/(3)e^(x)-2e^(-2x) Step 2 We have to verify the following differential equation: y'+2y=2e^(x) Substitute the expressions for y' and y, To get [(2)/(3)e^(x)-2e^(-2x)]+2[(2)/(3)e^(x)+e^(-2x)]=2e^(x) [(2)/(3)e^(x)-2e^(-2x)]+[(4)/(3)e^(x)+2e^(-2x)]=2e^(x) (2)/(3)e^(x)-2e^(-2x)+(4)/(3)e^(x)+2e^(-2x)=2e^(x) ((2)/(3)+(4)/(3))*e^(x)=2e^(x) Hence verified Result: Hint: y'=(2)/(3)e^(x)-2e^(-2x)
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