From 432dbec49f5f008c78b59fb47e8de28e77d67422 Mon Sep 17 00:00:00 2001 From: Rasmus Rosendahl-Kaa Date: Tue, 2 Dec 2025 11:46:46 +0100 Subject: [PATCH] =?UTF-8?q?Andenordens=20line=C3=A6re=20differentiallignin?= =?UTF-8?q?ger?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../Andenordens differentialligninger.pdf | 4234 +++++++++++++++++ .../Andenordens differentialligninger.typ | 262 + 2 files changed, 4496 insertions(+) create mode 100644 01001 - Matematik 1a/Noter/Andenordens differentialligninger/Andenordens differentialligninger.pdf create mode 100644 01001 - Matematik 1a/Noter/Andenordens differentialligninger/Andenordens differentialligninger.typ diff --git a/01001 - Matematik 1a/Noter/Andenordens differentialligninger/Andenordens differentialligninger.pdf b/01001 - Matematik 1a/Noter/Andenordens differentialligninger/Andenordens differentialligninger.pdf new file mode 100644 index 0000000..134f85c --- /dev/null +++ b/01001 - 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+ /Pg 278 0 R +>> +endobj + +97 0 obj +<< + /Type /StructElem + /S /P + /P 14 0 R + /K [450] + /Pg 278 0 R +>> +endobj + +98 0 obj +<< + /Type /StructElem + /S /P + /P 14 0 R + /K [417 99 0 R] + /Pg 278 0 R +>> +endobj + +99 0 obj +<< + /Type /StructElem + /S /Formula + /P 98 0 R + /K [418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449] + /Pg 278 0 R +>> +endobj + +100 0 obj +<< + /Type /StructElem + /S /P + /P 14 0 R + /K [101 0 R] +>> +endobj + +101 0 obj +<< + /Type /StructElem + /S /Formula + /P 100 0 R + /K [405 406 407 408 409 410 411 412 413 414 415 416] + /Pg 278 0 R +>> +endobj + +102 0 obj +<< + /Type /StructElem + /S /P + /P 14 0 R + /K [103 0 R] +>> +endobj + +103 0 obj +<< + /Type /StructElem + /S /Formula + /P 102 0 R + /K [393 394 395 396 397 398 399 400 401 402 403 404] + /Pg 278 0 R +>> +endobj + +104 0 obj +<< + /Type /StructElem + /S /P + /P 14 0 R + /K [384 105 0 R 392] + /Pg 278 0 R +>> 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date: datetime(year: 2025, month: 12, day: 02), + author: "Rasmus Rosendahl-kaa", + semester: "Spring 2025" +) + +#definition(title: "13.3.1")[ +$ +f''(t) + a_1 f'(t)+a_0 f(t)=q(t)\ +a_0,a_1 in RR, q(t) in C_infinity(RR) +$ +Kaldes en lineær andenordens differentialligning + +Er $q(t)=0$, kaldes ligningen *homogent*, ellers *inhomogent* +] + +#theorem(title: "Sætning 13.3.1")[ +$f(t)$ er løsning til $f''(t) + a_1 f'(t)+a_0 f(t)=q(t)$ hvis og kun hvis $vec(f(t),f'(t))$ er løsning til systemet: +$ +vec(f'_1(t),f'_2(t)) = mat(0,1;-a_0,-a_1) dot vec(f_1(t),f_2(t)) + vec(0,q(t)) +$ +] + +== Løsninger til det homogene system +Skriv $f_1(t)=f(t)$ og $f_2(t)=f'(t)$. Så fås: +$ f'_1(t)=f'(t)=f_2(t) $ +$ +f'_2(t)=f''(t)&=-a_1f'(t)-a_0f(t)+q(t)\ +&=-a_1f_2(t)-a_0f_1(t)+q(t) +$ +$ +vec(f'_1(t),f'_2(t))=mat(0,1;-a_0,-a_1) dot vec(f_1(t),f_2(t))+vec(0,q(t)) +$ + +For nu: antag $q(t)=0$. Dvs vi er i det homogene tilfælde + +$bold(A)=mat(0,1,-a_0;-a_1)$ + +/ 1): Undersøg $bold(A)$'s egenværdier. +$ P_bold(A)(z)=det(mat(-z,1;-a_0,-_1-z))&=-z dot (-a_1-z) - 1 dot (-a_0)\ +&= z^2+a_1z+a_0 +$ +Rødder findes i $P_bold(A)(z)$.\ +Diskriminant: $D=a_1^2-4a_0$ + +Rødder: $lambda_1 = (-a_1+sqrt(D))/2, lambda_2=(-a_1-sqrt(D))/2$ + +/ 2): Egenrummene: $lambda$ er en egenværdi +$ +E_lambda&=ker(bold(A)-lambda dot bold(I)_2)\ +&=ker(mat(-lambda,1;-a_0,-a_1-lambda)) +$ + +Vektoren $mat(-lambda,1;-a_0,-a_1-lambda)$ må have rang 1 da $lambda$ er en egenværdi og hvis rang var 2, ville kernen være nulvektoren. Dvs at vi må have mindst én nulrække: +$ mat(-lambda,1;-a_0,-a_1-lambda) arrow mat(-lambda,1;0,0) $ +$ bold(v) in E_lambda, bold(v)=vec(a,b) <=>\ +-lambda dot a + b = 0\ +E_lambda = "span"(vec(1,lambda)) +$ + +Nu skælner vi mellem 3 tilfælde: +/ 1): $D>0$: $bold(A)$ har to forskellige egenværdier i $RR$ +Fra sidste uge: har systemet $f'(t)=bold(A) dot f(t)$ fuldstændige løsning +$ +vec(f_1(t),f_2(t))=c_1 dot vec(1,lambda_1) dot e^(lambda_1 t) + c_2 dot vec(1,lambda_2) dot e^(lambda_2 t), (c_1,c_2 in RR)\ +lambda_1 = (-a_1+sqrt(D))/2, lambda_2=(-a_1-sqrt(D))/2 +$ +Derfor har differentialligningen $f''(t)+a_1 dot f'(t) + a_0 dot f(t)=0$ har fuldstændige løsning ($f(t)=f_1(t))$: +$ +f(t)=c_1 dot e^(lambda_1 t) + c_2 dot e^(lambda_2 t), (c_1,c_2 in RR) +$ + +#note-box()[ +Skal man i en opgave finde den fuldstændige løsning, hvor $a_0, a_1$ er givet, kan man strakt skrive det karakteristiske polynomium ned: $P_bold(A)(z) = z^2 + a_1 z + a_0$ +] + +/ 2): $D<0$. $sqrt(D)=i dot sqrt(|D|)$. +$ +lambda_1=(-a_1+i sqrt(|D|))/2, lambda_2 = (-a_1-i sqrt(|D|))/2\ +(overline(lambda_1)=lambda_2) +$ + +Fr sidste lille dag har systemet $bold(f'(t)=bold(A) dot bold(f(t)))$ har fuldstændige løsning: +$ +vec(f_1(t),f_2(t)) = c_1 dot "Re"(vec(1,lambda_1) dot e^(lambda_1 t)) + c_2 dot "Im"(vec(1,lambda_1) dot e^(lambda_1 t))\ +(c_1, c_2 in RR) +$ +Vi er ligeglad med $f_2(t)$. Vi ved: $lambda_1 = (-a_1)/2 + i dot (sqrt(|D|))/2 = alpha+ i beta, (alpha = -a_1/2, beta = sqrt(|D|)/2)$ +$ +e^(lambda_1 t) = e^((alpha + i beta)t)&=e^(alpha t + i beta t)=e^(alpha t) dot e^(i beta t)\ +&= e^(alpha t) dot (cos(beta t) + i dot sin(beta t))\ +&= e^(alpha t) cos(beta t) + i e^(alpha t) dot sin(beta t) +$ + +Derfor har $f''(t)+a_1 dot f'(t) + a_0 f(t)=0$ fuldstændige løsning +$ +f(t)=c_1 dot e^(alpha t) dot cos(beta t) + c_2 dot e^(alpha t) dot sin(beta t)\ +(c_1, c_2 in RR) +$ + +/ 3): $D=0$. $lambda_1 = lambda_2 = (-a_1 plus.minus sqrt(0))/2$. $"am"(lambda_1) = 2$. + +$E_lambda_1 = "span"(vec(1, lambda_1))$ dvs $"gm"(lambda_1) = 1$ + +Én løsning er $f(t)=e^(lambda_1 t)$ + +Idéen for at finde den fuldstændige løsning til $f''(t)+a_1 dot f'(t) + a_0 dot f(t)=0$ hvis $D=0$ + +Vi ved $lambda_1^2 + a_1 lambda_1 + a_0 = 0$ og $2 lambda_1 + a_1 = 0$ + +Prøv at find en løsning på formen: +$ +f(t)=g(t) dot e^(lambda_1 t) +$ +Så gælder: +$ +f'(t)=g'(t) dot e^(lambda_1 t) + g(t) dot lambda_1 dot e^(lambda_1 t)\ +$ og $ +f''(t)=g''(t) dot e^(lambda_1 t) + g'(t) dot lambda_1 e^(lambda_1 t) + g'(t) dot lambda_1 e^(lambda_1 t) + g(t) dot lambda_1^2 e^(lambda_1 t) +$ + +Indsættes udtrykkene for $f, f', f''$ i $f''(t)+a_1 dot f'(t) + a_0 dot f(t)=0$ fås: +$ +e^(lambda_1 t) dot g''(t) = 0 +$ +Da $e^(lambda_1 t)$ ikke kan være 0, fås: $g''(t) = 0$. Dsv: +$ +g'(t) = c_1 +$ + +$g(t) = c_1 dot t + c_2, (c_1, c_2 in RR)$ + +Husk $f(t)=g(t) dot e^abs((lambda_1 t))$. + +Derfor: Hvis $D=0$, så har $f''(t)+a_1 dot f'(t) + a_0 dot f(t)=0$ den fuldstændige løsning: + +$ +f(t)=c_1 dot t dot e^(lambda_1 t)+c_2 dot e^(lambda_1 t), (c_1, c_2 in RR) +$ + + +#example()[ +Find fuldstændige løsninger for $f''(t)+4f'(t)+4 f(t)=0$ + +#solution()[ +Det karakteristiske polynomium er $z^2+4z+4$. + +$D = 4^2 - 4 dot 1 dot 4 = 0$ + +$lambda_1 = lambda_2 = -4/2 = -2$ + +Den fuldstændige løsning: $f(t)=c_1 dot t dot e^(-2 t) + c_2 dot e^(-2 t), (c_1, c_2 in RR)$ +]] + +#example()[ +Find fuldstændige løsninger for $f''(t)+4f'(t)+13f(t)=0$ + +#solution()[ +$D=4^2-4 dot 13 = -36$ + +$P_bold(A)(z)=z^2+4z+13$ + +$lambda_1 = (-4+6 i)/2 = -2+3i$ + +Den fuldstændige løsning: +$f(t)=c_1 dot e^(-2t) dot cos(3t) + c_2 e^(-2t) dot sin(3t), (c_1, c_2 in RR)$ +]] + +== Løsninger til det inhomogene system +#corollary(name: "13.3.2")[ +$ f''(t)+a_1f'(t)+a_0f(t)=q(t), (q(t) eq.not 0) $ +Lad $f_p (t)$ være en partikulær løsning til den givne differentialligning, så er den fuldstændige løsning: +$ +f(t)=f_p (t) + f_h (t) +$ +hvor $f_h (t)$ er den fuldstændige løsning til $f''(t)+a_1 f'(t)+a_0 f(t)=0$ +] + +#example()[ +Find den fuldstændige løsning til: +$ f''(t)+4f'(t)+4f(t)=cos(2t) $ + +#solution()[ +Vi kender den fuldstændige løsning til den homogene ligning,$f_h (t)$, fra tidligere eksempel: $f_h (t)=c_1 dot t dot e^(-2t) + c_2 e^(-2t)$ + +Vi skal finde en partikulær løsning. Man skal "gætte" på strukturen af en partikulær løsning + +Et gæt kunne være: $f_p (t)=a dot cos(2t)$. Ville ikke være rigtigt fordi $cos(2t)' = -2sin(2t)$, som vi ikke kan komme af med i vores ligning (som skal være lig med $cos(2t)$). + +Et andet gæt kunne være: $f_p (t)=a dot cos(2t) + b dot sin(2t)$. Så fås: +$ +f'_p (t)=a dot -sin(2t) dot 2 + b dot cos(2t) dot 2\ +f''_p (t)=a dot -cos(2t) dot 4 + b dot -sin(2t) dot 4 +$ +Vi skal nu se om vi kan finde $a,b$ så alt hænger sammen. + +Indsætter det i ligningen $f''(t)+4f'(t)+4f(t)=cos(2t)$: +$ +underbrace(-4a dot cos(2t)-4b dot sin(2t), f''(t)) + 4 dot underbrace((-2a dot sin(2t) + 2b dot cos(2t)), f'(t))\ ++ 4 dot underbrace((a cos(2t) + b sin(25)), f(t)) = cos(2t) +$ +Kan samle alt med $cos(2t)$ sammen, som skal give $cos(2t)$ og alt med $sin(2t)$ sammen, som skal give 0. +$ +(-4a+8b+4a) dot cos(2t) + (-4b-8a+4b) dot sin(2t)=cos(2t)\ +8b dot cos(2t) - 8a dot sin(2t) = cos(2t)\ +8b = 1, b = 1/8\ +8a=0, a = 0 +$ +Dvs. $f_p (t) = 0 dot cos(2t) + 1/8 dot sin(2t)$ er en partikulær løsning. + +$ +f(t)=1/8 sin(2t) + c_1 t e^(-2t) + c_2 e^(-2t)\ +(c_1, c_2 in RR) +$ +Er den ønskede fuldstændige løsning +]] + +== Begyndelsesbetingelser/Begyndelsesværdibetingelser +$f''(t)+a_1f'(t)+a_0f(t)=q(t)$ + +$vec(f(t),f'(t))$ er en løsning til $vec(f'_1 (t), f'_2 (t))=mat(0,1;-a_0,-a_1) dot vec(f_1 (t), f_2 (t)) + vec(0,q(t))$ + +$vec(0,q(t))$ $<-$ Begyndelsesværdibetingelse: $cases(f_1 (t_0)=y_0, f_2 (t_0) = z_0)$ + + +Da $f_1 (t) = f(t)$ og $f_2(t) = f'(t)$ er begyndelsesværdibetingelse: $cases(f(t_0) = y_0,f'(t_0)=z_0)$ + +Ligesom til systemer, kan man hvis man har den fuldstændige løsning bestemme de ubekendte vha. begyndelsesværdibetingelserne. + +#example(title: "fortsat")[ +Find $f(t) in C_infinity (RR)$ som opfylder differentialligningen $f''(t)+4f'(t)+4f(t)=cos(2t)$ og begyndelsesbetingelsen $f(0)=1)$ og $f'(0)=0$. + +#solution()[ +Vi kender den fuldstændige løsning: +$ f(t)=c_1 t e^(-2t) + c_2 e^(-2t) + 1/8 sin(2t) $ +$ +f(0)=1 <=> c_2 = 0 +$ +Før $f'(0)$ kan regnes, skal $f'(t)$ findes +$ +f'(t)=c_1 dot (1 dot e^(-2t)+ t dot -2e^(-2t) + t dot -2e^2t)\ ++c_2 dot (-2 dot e^(-2t)) + 1/8 dot cos(2t) dot 2 +$ +$ +f'(0)=0 <=> c_1 + c_2 dot -2 + 1/8 dot 2 = 0 +$ +Derfor: $c_1 = 2- 1/4 = 7/4$ + +Den ønskede løsning er: +$ +f(t) = t e^(-2t) + 7/4 e^(-2t) + 1/8 sin(2t) +$ +]] +