Question
Using the method of variation of parameters, solve the differential equation:
Answer :
Word Count : 325
The given differential equation is: \[ \frac{\mathrm{d^{2}y}}{\mathrm{d} x^{2}} + y = \sec^3 x \] We will solve this using the method of variation of parameters. Let's break it down: ### Step 1: Solve the homogeneous equation First, solve the corresponding homogeneous equation: \[ \frac{\mathrm{d^{2}y}}{\mathrm{d} x^{2}} + y = 0 \] The characteristic equation for this is: \[ r^2 + 1 = 0 \] This gives complex roots \(r = \pm i\). Hence, the solution to the homogeneous equation is: \[ y_h(x) __________ _________ ________ __________ ___ ________ __________ ____ _________.
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The given differential equation is: \[ \frac{\mathrm{d^{2}y}}{\mathrm{d} x^{2}} + y = \sec^3 x \] We will solve this using the method of variation of parameters. Let's break it down: ### Step 1: Solve the homogeneous equation First, solve the corresponding homogeneous equation: \[ \frac{\mathrm{d^{2}y}}{\mathrm{d} x^{2}} + y = 0 \] The characteristic equation for this is: \[ r^2 + 1 = 0 \] This gives complex roots \(r = \pm i\). Hence, the solution to the homogeneous equation is: \[ y_h(x) __________ _________ ________ __________ ___ ________ __________ ____ _________.
__________ _________ ___ ______ _____ _______ ________ ____ ___ ____ _____.
______ _____ ________ __________ ______ _____ __________ ________ ___ __________ _________ _____.
_____ ________ _______ _______ ________ _____ ____ ____ _________ __________.
_________ _________ ________ ________ _____ _________ _________ ___.
____ ___ ________ _________ ___ ____ ________ __________ ______ ________.
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______ ____ _______ ________ ___ ________.
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__________ ____ __________ ________ _______ ________ _____ ___ _______.
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________ ___.
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