Eritque Arcus Math Notes

order reduce by substitution

1 min read

a technique to solve higher order differential equation by setting additional variable to substitute. Then solve it by normal ODE method. Lastly substitute it back

Example

(2−𝑡)𝑢‴+(3−𝑡)𝑢″=0

where 𝑡<2
consider 𝑧=𝑢″

(2−𝑡)𝑧′+(3−𝑡)𝑧=0𝑧′=𝑡−32−𝑡𝑧𝑧′1𝑧=𝑡−32−𝑡𝑧=𝐶𝑒∫𝑡−32−𝑡𝑑𝑡𝑧=𝐶(𝑡−2)𝑒−𝑡

then

𝑢″=𝐶(𝑡−2)𝑒−𝑡𝑢′=−𝐶1𝑒−𝑡(𝑡−1)+𝐶2𝑢=−𝐶1𝑒−𝑡𝑡+𝐶3+𝐶2𝑡

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