{"id":9356,"date":"2025-05-20T18:57:40","date_gmt":"2025-05-20T18:57:40","guid":{"rendered":"https:\/\/presswiki.allmath.gr\/wpwiki18\/?p=9356"},"modified":"2025-05-20T19:20:14","modified_gmt":"2025-05-20T19:20:14","slug":"%ce%bc%ce%af%ce%b1-%ce%b4%ce%b9%ce%b1%cf%86%ce%bf%cf%81%ce%b9%ce%ba%ce%ae-%ce%b5%ce%be%ce%af%cf%83%cf%89%cf%83%ce%b7","status":"publish","type":"post","link":"https:\/\/presswiki.allmath.gr\/wpwiki18\/2025\/05\/20\/18\/57\/40\/9356\/","title":{"rendered":"\u039c\u03af\u03b1 \u03b4\u03b9\u03b1\u03c6\u03bf\u03c1\u03b9\u03ba\u03ae \u03b5\u03be\u03af\u03c3\u03c9\u03c3\u03b7 20-05-2025:"},"content":{"rendered":"\n<ul><li>\u039d\u03b1 \u03b2\u03c1\u03b5\u03b8\u03b5\u03af \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 $f$, \u03ce\u03c3\u03c4\u03b5:<\/li><li>$$f:(-1, + \\infty) \\rightarrow (-1 , +\\infty)$$, \u03ba\u03b1\u03b9<\/li><li>$$ (x+1)f'(x)-x(f(x) + 1) , f(0)= 0 $$<\/li><\/ul>\n\n\n\n<!--more-->\n\n\n\n<p>\u03a5\u03c0\u03cc\u03b4\u03b5\u03b9\u03be\u03b7: \u0395\u03c6\u03cc\u03c3\u03bf\u03bd \u03c7>-1 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 \u03b9\u03c3\u03bf\u03b4\u03cd\u03bd\u03b1\u03bc\u03b1 \u03cc\u03c4\u03b9: <\/p>\n\n\n\n<p>$$f'(x) = \\frac{x}{x+1} (f(x) + 1) $$<\/p>\n\n\n\n<p>\u0398\u03ad\u03c4\u03bf\u03c5\u03bc\u03b5 $$h(x) = f(x) + 1 \\Rightarrow h'(x) = f'(x)$$ \u03ba\u03b1\u03b9 \u03b7 \u03b5\u03be\u03af\u03c3\u03c9\u03c3\u03b7 \u03b3\u03af\u03bd\u03b5\u03c4\u03b1\u03b9: <\/p>\n\n\n\n<p>$$h'(x) = \\frac{x}{x+1} h(x) \\Leftrightarrow$$<\/p>\n\n\n\n<p>$$h'(x) -\\frac{x}{x+1} h(x) = 0 $$<\/p>\n\n\n\n<p>\u03c0\u03bf\u03c5 \u03b8\u03c5\u03bc\u03af\u03b6\u03b5\u03b9 \u03c0\u03bf\u03bb\u03bb\u03b1\u03c0\u03bb\u03b1\u03c3\u03b9\u03b1\u03c3\u03c4\u03ae Euler&#8230;<\/p>\n\n\n\n<p>\u03a0\u03c1\u03b5\u03c0\u03b5\u03b9 \u03bd\u03b1 \u03b2\u03c1\u03bf\u03cd\u03bc\u03b5 \u03b1\u03c1\u03c7\u03b9\u03ba\u03ae \u03c4\u03b7\u03c2 $$-\\frac{x}{x+1}$$ \u03ba\u03b1\u03b9 \u03bd\u03b1 \u03c0\u03bf\u03bb\u03bb\u03b1\u03c0\u03bb\u03b1\u03c3\u03b9\u03ac\u03c3\u03bf\u03c5\u03bc\u03b5 \u03bc\u03b5 $$e^{\\int A(x) }$$<\/p>\n\n\n\n<p>\u0392\u03c1\u03af\u03c3\u03ba\u03bf\u03c5\u03bc\u03b5 \u03bc\u03af\u03b1 \u03b1\u03c1\u03c7\u03b9\u03ba\u03ae \u03b3\u03b9\u03b1 \u03c4\u03b7\u03bd $$ \\int -\\frac{x}{x+1} =  \\int -\\frac{x+1-1}{x+1} =- \\int 1-\\frac{1}{x+1} = -x + ln|x+1|$$ <\/p>\n\n\n\n<p>\u039f\u03c0\u03cc\u03c4\u03b5 \u03c0\u03bf\u03bb\u03bb\u03b1\u03c0\u03bb\u03b1\u03c3\u03b9\u03ac\u03b6\u03bf\u03c5\u03bc\u03b5 \u03bc\u03b5 $$e^{ -x + ln(x+1)}$$ \u03ba\u03b1\u03b9 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5: <\/p>\n\n\n\n<p>$$e^{ -x + ln(x+1) } h'(x) -e^{ -x + ln(x+1)} \\frac{x}{x+1} h(x) = 0 \\Leftrightarrow$$<\/p>\n\n\n\n<p>$$e^{ -x + ln(x+1)} h'(x)  + (e^{ -x + ln(x+1)})&#8217; h(x) = 0 $$<\/p>\n\n\n\n<p>$$(e^{ -x + ln(x+1)} h(x))&#8217; = 0 $$<\/p>\n\n\n\n<p>$$e^{ -x + ln(x+1)} h(x) = c $$<\/p>\n\n\n\n<p>$$ h(x) = c \\cdot e^{ -x + ln(x+1)} $$<\/p>\n\n\n\n<p>$$ h(x) = c  (\\cdot e^{ -x}  +x+1 )$$<\/p>\n\n\n\n<p>\u0391\u03c0&#8217; \u03cc\u03c0\u03bf\u03c5 \u03c0\u03c1\u03bf\u03ba\u03cd\u03c0\u03c4\u03b5\u03b9 \u03b5\u03cd\u03ba\u03bf\u03bb\u03b1 \u03b7 \\(f(x)\\) .<\/p>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u039d\u03b1 \u03b2\u03c1\u03b5\u03b8\u03b5\u03af \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 $f$, \u03ce\u03c3\u03c4\u03b5: $$f:(-1, + \\infty) \\rightarrow (-1 , +\\infty)$$, \u03ba\u03b1\u03b9 $$ (x+1)f'(x)-x(f(x) + 1) , f(0)= 0 $$<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[9,3,15],"tags":[935],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/presswiki.allmath.gr\/wpwiki18\/wp-json\/wp\/v2\/posts\/9356"}],"collection":[{"href":"https:\/\/presswiki.allmath.gr\/wpwiki18\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/presswiki.allmath.gr\/wpwiki18\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/presswiki.allmath.gr\/wpwiki18\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/presswiki.allmath.gr\/wpwiki18\/wp-json\/wp\/v2\/comments?post=9356"}],"version-history":[{"count":13,"href":"https:\/\/presswiki.allmath.gr\/wpwiki18\/wp-json\/wp\/v2\/posts\/9356\/revisions"}],"predecessor-version":[{"id":9376,"href":"https:\/\/presswiki.allmath.gr\/wpwiki18\/wp-json\/wp\/v2\/posts\/9356\/revisions\/9376"}],"wp:attachment":[{"href":"https:\/\/presswiki.allmath.gr\/wpwiki18\/wp-json\/wp\/v2\/media?parent=9356"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/presswiki.allmath.gr\/wpwiki18\/wp-json\/wp\/v2\/categories?post=9356"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/presswiki.allmath.gr\/wpwiki18\/wp-json\/wp\/v2\/tags?post=9356"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}