MAT244--2019F > Quiz-3

QUIZ3 TUT 0502

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Xinyu Jing:
Question: 𝑐𝑜𝑠(𝑡)𝑦″+𝑠𝑖𝑛(𝑡)𝑦′−𝑡𝑦=0
Find the Wronskian of two solutions of the given differential equation without solving the equation.

Solution:
Divide both sides by 𝑐𝑜𝑠(𝑡)
𝑦″+𝑡𝑎𝑛(𝑡)𝑦′−𝑡𝑐𝑜𝑠(𝑡)𝑦=0
𝑊(𝑦1,𝑦2)(𝑡)=𝑐𝑒−∫𝑝(𝑡)𝑑𝑡
𝑊(𝑦1,𝑦2)(𝑡)=𝑐𝑒−∫𝑡𝑎𝑛(𝑡)𝑑𝑡=𝑐𝑒−(−𝑙𝑛|𝑐𝑜𝑠(𝑡)|)
𝑊(𝑦1,𝑦2)(𝑡)=𝑐𝑒𝑙𝑛|𝑐𝑜𝑠(𝑡)|=𝑐𝑐𝑜𝑠(𝑡)

Therefore, the Wronskian of any pair of solutions of the given equation is 𝑊(𝑦1,𝑦2)(𝑡)=𝑐𝑐𝑜𝑠(𝑡)