Undetermined Coefficients We'll solve some second order equations using the method of undetermined coefficients. We'll use the computer to do the algebra. First Equation: QyQ+SSRlcTFHNiIvLCYtSSVEaWZmR0YlNiUtSSJ5R0YlNiNJInRHRiVGLkYuIiIiRisiIiUsJkYuRi8qJkYuRi8tSSRzaW5HRiU2IywkRi4iIiNGL0YvRi8= LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LywmLUklRGlmZkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYlLUkieUdGKTYjSSJ0R0YpRi5GLiIiIkYrIiIlLCZGLkYvKiZGLkYvLUkkc2luR0YmNiMsJEYuIiIjRi9GLw== JSFH The characteristic equation is: QyQ+SSZjaGFyMUc2Ii8sJiokSSJyR0YlIiIjIiIiIiIlRisiIiFGKw== LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LywmKiRJInJHNiIiIiMiIiIiIiVGKCIiIQ== QyQtSSZzb2x2ZUc2IjYkSSZjaGFyMUdGJUkickdGJSIiIg== LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder NiReIyIiI14jISIj JSFH The general solution to the homogeneous equation is QyQ+SSR5aDFHNiIsJiomSSNjMUdGJSIiIi1JJGNvc0dGJTYjLCRJInRHRiUiIiNGKUYpKiZJI2MyR0YlRiktSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiVGLEYpRilGKQ== LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCYqJkkjYzFHNiIiIiItSSRjb3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywkSSJ0R0YlIiIjRiZGJiomSSNjMkdGJUYmLUkkc2luR0YpRixGJkYm JSFH The trial solution for the particular solution is: QyQ+SSV5cHQxRzYiLCoqJkkjZDFHRiUiIiJJInRHRiVGKUYpSSNkMkdGJUYpKiZGKkYpLCYqKEkjZDNHRiVGKUYqRiktSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywkRioiIiNGKUYpKiZJI2Q0R0YlRilGMEYpRilGKUYpKiZGKkYpLCYqKEkjZDVHRiVGKUYqRiktSSRjb3NHRjJGNUYpRikqJkkjZDZHRiVGKUY+RilGKUYpRilGKQ== LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCoqJkkjZDFHNiIiIiJJInRHRiVGJkYmSSNkMkdGJUYmKiZGJ0YmLCYqKEkjZDNHRiVGJkYnRiYtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywkRiciIiNGJkYmKiZJI2Q0R0YlRiZGLUYmRiZGJkYmKiZGJ0YmLCYqKEkjZDVHRiVGJkYnRiYtSSRjb3NHRi9GMkYmRiYqJkkjZDZHRiVGJkY7RiZGJkYmRiY= QyQ+SSNsMUc2Ii1JKXNpbXBsaWZ5R0YlNiMsJi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiVJJXlwdDFHRiVJInRHRiVGLyIiIkYuIiIlRjA= LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LDIqJkkjZDNHNiIiIiItSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywkSSJ0R0YlIiIjRiZGLyooRiRGJkYuRiYtSSRjb3NHRilGLEYmIiIpKiZJI2Q0R0YlRiZGMUYmIiIlKiZJI2Q1R0YlRiZGMUYmRi8qKEY4RiZGLkYmRidGJiEiKSomSSNkNkdGJUYmRidGJiEiJSomSSNkMUdGJUYmRi5GJkY2SSNkMkdGJUY2 QyQ+SSVwZXExRzYiLUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2Iy8sKEkjbDFHRiUiIiJJInRHRiUhIiIqJkYwRi8tSSRzaW5HRig2IywkRjAiIiNGL0YxIiIhRi8= LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder Lyw2KiZJI2QzRzYiIiIiLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiMsJEkidEdGJiIiI0YnRjAqKEYlRidGL0YnLUkkY29zR0YqRi1GJyIiKSomSSNkNEdGJkYnRjJGJyIiJSomSSNkNUdGJkYnRjJGJ0YwKihGOUYnRi9GJ0YoRichIikqJkkjZDZHRiZGJ0YoRichIiUqJkkjZDFHRiZGJ0YvRidGN0kjZDJHRiZGN0YvISIiKiZGL0YnRihGJ0ZCIiIh JSFH JSFH We collect all the coefficients and then solve the linear system. JSFH QyQ+SSR0ZTFHNiItSShjb2xsZWN0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiRJJXBlcTFHRiUtSSRjb3NHRig2IywkSSJ0R0YlIiIjIiIi LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LywyKiYsKEkjZDRHNiIiIiVJI2Q1R0YnIiIjKiZJI2QzR0YnIiIiSSJ0R0YnRi0iIilGLS1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJzYjLCRGLkYqRi1GLSomRixGLS1JJHNpbkdGMkY1Ri1GKiomSSNkNkdGJ0YtRjhGLSEiJSomSSNkMUdGJ0YtRi5GLUYoSSNkMkdGJ0YoKihGKUYtRi5GLUY4Ri0hIikqJkYuRi1GOEYtISIiRi5GQyIiIQ== QyQ+SSVsZXExRzYiLywkSSNkM0dGJSIiKSIiISIiIg== LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LywkSSNkM0c2IiIiKSIiIQ== QyQ+SSVsZXEyRzYiLywmSSNkNEdGJSIiJUkjZDVHRiUiIiMiIiEiIiI= LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LywmSSNkNEc2IiIiJUkjZDVHRiUiIiMiIiE= QyQ+SSR0ZTJHNiItSShjb2xsZWN0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiRJJXBlcTFHRiUtSSRzaW5HRig2IywkSSJ0R0YlIiIjIiIi LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LywwKiYsKkkjZDNHNiIiIiNJI2Q2R0YnISIlKiZJI2Q1R0YnIiIiSSJ0R0YnRi0hIilGLiEiIkYtLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YnNiMsJEYuRihGLUYtRi5GMCooRiZGLUYuRi0tSSRjb3NHRjNGNkYtIiIpKiZJI2Q0R0YnRi1GOUYtIiIlKiZGLEYtRjlGLUYoKiZJI2QxR0YnRi1GLkYtRj5JI2QyR0YnRj4iIiE= QyQ+SSVsZXEzRzYiLywmSSNkNUdGJSEiKSEiIiIiIiIiIUYr LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LywmSSNkNUc2IiEiKSEiIiIiIiIiIQ== QyQ+SSVsZXE0RzYiLywmSSNkM0dGJSIiI0kjZDZHRiUhIiUiIiEiIiI= LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LywmSSNkM0c2IiIiI0kjZDZHRiUhIiUiIiE= QyQ+SSR0ZTNHNiItSShjb2xsZWN0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiRJJXBlcTFHRiU8JC1JJGNvc0dGKDYjLCRJInRHRiUiIiMtSSRzaW5HRihGMCIiIg== LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LywsKiYsKEkjZDRHNiIiIiVJI2Q1R0YnIiIjKiZJI2QzR0YnIiIiSSJ0R0YnRi0iIilGLS1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJzYjLCRGLkYqRi1GLSomLCpGLEYqSSNkNkdGJyEiJSomRilGLUYuRi0hIilGLiEiIkYtLUkkc2luR0YyRjVGLUYtKiZJI2QxR0YnRi1GLkYtRihJI2QyR0YnRihGLkY9IiIh QyQ+SSVsZXE1RzYiLywmSSNkMUdGJSIiJSEiIiIiIiIiIUYr LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LywmSSNkMUc2IiIiJSEiIiIiIiIiIQ== QyQ+SSVsZXE2RzYiLywkSSNkMkdGJSIiJSIiISIiIg== LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LywkSSNkMkc2IiIiJSIiIQ== QyQ+SSNsc0c2Ii1JJnNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiQ8KEklbGVxMUdGJUklbGVxMkdGJUklbGVxM0dGJUklbGVxNEdGJUklbGVxNUdGJUklbGVxNkdGJTwoSSNkMUdGJUkjZDJHRiVJI2QzR0YlSSNkNEdGJUkjZDVHRiVJI2Q2R0YlIiIi LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder PCgvSSNkMUc2IiMiIiIiIiUvSSNkMkdGJSIiIS9JI2QzR0YlRisvSSNkNEdGJSNGJyIjOy9JI2Q1R0YlIyEiIiIiKS9JI2Q2R0YlRis= QyQ+SSR5cDFHNiItSSVzdWJzRyUqcHJvdGVjdGVkRzYkSSNsc0dGJUkleXB0MUdGJSIiIg== LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LChJInRHNiIjIiIiIiIlKiZGI0YmLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMsJEYjIiIjRiYjRiYiIzsqJkYjRjAtSSRjb3NHRitGLkYmIyEiIiIiKQ== QyQ+SSN5MUc2IiwmSSR5aDFHRiUiIiJJJHlwMUdGJUYoRig= LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCwqJkkjYzFHNiIiIiItSSRjb3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywkSSJ0R0YlIiIjRiZGJiomSSNjMkdGJUYmLUkkc2luR0YpRixGJkYmRi4jRiYiIiUqJkYuRiZGMkYmI0YmIiM7KiZGLkYvRidGJiMhIiIiIik= JSFH JSFH QyQtSSdkc29sdmVHNiI2I0kkZXExR0YlIiIi LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder Ly1JInlHNiI2I0kidEdGJSwuKiYtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywkRiciIiMiIiJJJF9DMkdGJUYyRjIqJi1JJGNvc0dGLEYvRjJJJF9DMUdGJUYyRjJGJyNGMiIiJUY1I0YyIiNLKiZGJ0YyRipGMiNGMiIjOyomRidGMUY1RjIjISIiIiIp JSFH JSFH JSFH JSFH JSFH Second Equation QyQ+SSRlcTJHNiIvLCgtSSVkaWZmRyUqcHJvdGVjdGVkRzYlLUkieUdGJTYjSSJ0R0YlRi9GLyIiIi1GKTYkRixGLyIiJEYsIiIjKiZGL0YwLUkkZXhwRzYkRipJKF9zeXNsaWJHRiU2IywkRi8hIiJGMEYw LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); 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# input placeholder LCYqJkkjYzFHNiIiIiItSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywkSSJ0R0YlISIiRiZGJiomSSNjMkdGJUYmLUYoNiMsJEYuISIjRiZGJg== JSFH We write down our trial solution. 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