MindMap Gallery differential equations
This figure summarizes the content of the differential equations chapter in Advanced Mathematics. The content is detailed and the steps are comprehensive. A differential equation containing the derivative or differential of an unknown function is called a differential equation. A differential equation in which the unknown function is a function of one variable is called an ordinary differential equation. The general form is f(x, y, y'...y(n))=0, and the standard form is y(n)=f(x, y, y'...y(n-1)).
Edited at 2023-05-01 17:57:13This is a mind map about bacteria, and its main contents include: overview, morphology, types, structure, reproduction, distribution, application, and expansion. The summary is comprehensive and meticulous, suitable as review materials.
This is a mind map about plant asexual reproduction, and its main contents include: concept, spore reproduction, vegetative reproduction, tissue culture, and buds. The summary is comprehensive and meticulous, suitable as review materials.
This is a mind map about the reproductive development of animals, and its main contents include: insects, frogs, birds, sexual reproduction, and asexual reproduction. The summary is comprehensive and meticulous, suitable as review materials.
This is a mind map about bacteria, and its main contents include: overview, morphology, types, structure, reproduction, distribution, application, and expansion. The summary is comprehensive and meticulous, suitable as review materials.
This is a mind map about plant asexual reproduction, and its main contents include: concept, spore reproduction, vegetative reproduction, tissue culture, and buds. The summary is comprehensive and meticulous, suitable as review materials.
This is a mind map about the reproductive development of animals, and its main contents include: insects, frogs, birds, sexual reproduction, and asexual reproduction. The summary is comprehensive and meticulous, suitable as review materials.
No relevant template
differential equations
basic concept
1. Definition: An equation containing the derivative or differential of an unknown function is called a differential equation, and a differential equation in which the unknown function is a function of one variable is called an ordinary differential equation. The general form is f(x, y, y'...y(n)) = 0, the standard form is y(n)=f(x, y, y'...y(n-1)).
2. Order: the highest order of the derivative or differential of the unknown function
3. Solution: function that satisfies the differential equation General solution: A solution that contains the same number of independent constants as the order of the equation Special solution: solution that does not contain arbitrary constants
4. Initial conditions: conditions for determining any constant in the general solution
theorem
1.y1(x), y2(x) are two second-order homogeneous (linearly independent) solutions, then any linear combination C1y1(x) C2y2(x) is also the (general) solution of this homogeneous differential equation .
2.y1(x), y2(x) are two second-order non-homogeneous linearly independent solutions. For any a b=1, ay1(x) by2(x) is also the solution of the non-homogeneous differential equation. For For any a b=0, ay1(x) by2(x) is the solution of the homogeneous equation corresponding to the equation. In the same way, three solutions are three coefficients.
3. If C1y1(x) C2y2(x) is the general solution of the second-order homogeneous equation, and y*(x) is the special solution of the second-order non-homogeneous equation, then C1y1(x) C2y2(x) y*(x) is not Homogeneous and convenient general solution. 【Feitong=Qitong Feite】
4. If y*1(x) is a special solution of the second-order non-homogeneous equation 1, and y*2(x) is a special solution of the second-order non-homogeneous equation 2, then y*1(x) y*2( x) is a special solution of the second-order non-homogeneous equation 1 and equation 2. [The principle of superposition of solutions]
5.y1, y2, and y3 are three non-homogeneous linearly independent solutions. The general solution is y=C1 (y2-y1) C2 (y3-y2) y3 [subtract any two, and finally add any one solution]
6. The difference between two non-homogeneous special solutions is a homogeneous special solution (used in 5)
1. First-order differential equation
1. Separable variables
y'=f(x)·g(y)
Solution: ∫dy/g(y)=∫f(x)dx
2. Homogeneous equations
y'=f(y/x)
solution: ①Let u=y/x, y'=u+xu' ②So u+xu′=f(u) ③∫du/[f(u)-u]=∫dx/x=lnIxl+C ④u restores u to y/x
3. Can be transformed into a homogeneous equation
4. First-order linear equation
y'+P(x)y=Q(x)
solution: 1. Homogeneous general solution: y=Ce^(-∫P(x)dx) 2. Non-homogeneous general solution: y=e^(-∫P(x)dx)[∫Q(x)e^(∫P(x)dx)+C]
5. Bernoulli’s equation
y+P(x)y=Q(x)y^n(n≠0,1)
solution: ①Let z=y^(1-n) ②Then z+(1-n)P(x)z=(1-n)Q(x) ③According to the 4 solution method
2. Reducible higher-order differential equations
1.y^n=f(x)
Solution: n times of integration
2.y"=f(x,y')
Solution: Let y'=p, turn into p'=f (x, p)
3.y"=f(y,y')
solution: ①Let y'=p, y"=pdp/dy ② Change to pdp/dy=f(y,p) ③Separate the variables after deriving the derivative of y...
3. Structure of solutions to linear differential equations
y"+P(x)y++q(x)y=f(x)① y"+P(x)y'+q(x)y=0 ②
1. If y1(x) and y2(x) are two linearly independent solutions of ②, then y=C1y1 C2y2 is the general solution of ②
2. If y* is a solution of ①, and y=C1y1 C2y2 is the general solution of ②, then y=y* C1y1 C2y2 is the general solution of ①
3.y1 is the solution of ①1, y2 is the solution of ①2, then y1 y2 is the solution of ①1 ①2
4. Solution to second-order constant coefficient linear differential equations
1.Solution steps
① Find the general solution Y of the corresponding homogeneous equation
② Find the special solution y* of the non-homogeneous equation
③Write the general solution y=Y y*
④Find the special solution according to the definite solution conditions
2. Find the general solution Y for y"+py'+qy=0
1. Write the characteristic equation r^2+pr+q=0 and find r1, r2
2. Write the general solution of the equation
①When r1≠r2, Y=C1e^(r1x)+C2e^(r2x)
②When r1=r2, Y=(C1+C2x)e^(r1x)
③When r1, 2=α±阝i, Y=e^(αx)〈C1cos阝x+C2sin阝x〉
3. Use the undetermined coefficient method to find the special solution y* of the non-homogeneous equation
y"+py+qy=
type
Pm(x)e^(αx)
e\(αx)〈Pl(x)cos阝x+Pm(x)sin阝x〉
1. Set y*= according to the equation
x^kQm(x)e^(αx)
x^ke^(αx)〈Rn(x)cos阝x+Tn(x)sin阝x〉
Among them, n=max(l,m), Qm(x) is a general polynomial of degree m of x, if Pm(x)=x^2, then Qm(x)=ax^2 bx c, Rn(x), Tn(x) is a general polynomial of degree n of x
2. The choice of k is divided into the following situations:
①When α or α 靝i is not a characteristic root, k=0
②When α or α 阝i is a single root of the characteristic equation, k=1
③When α is a multiple root of the characteristic equation, k=2
f(x)=Pn(x) Pn(x) is a polynomial of degree n
①0 is not a characteristic root: y*=Hn(x)
②0 is the characteristic single root: y*=xHn(x)
③0 is the characteristic multiple root: y*=x^2Hn(x)
f(x)=Pn(x)e^(αx)
①α is not a characteristic root: y*=Hn(x)e^(αx)
②α is a characteristic single root: y*=xHn(x)e^(αx)
③α is a characteristic multiple root: y*=x^2Hn(x)e^(αx)
f(x)=e^(αx)〈Pn(x)sin阝x+Qm(x)cos阝x〉
α靝i is not a characteristic root: y*=e^(αx)〈Rl(x)cos阝x+Sl(x)sin阝x〉
α靝i is the characteristic root: y*=xe^(αx)〈Rl(x)cos阝x+Sl(x)sin阝x〉
l=max(n,m)
5. Solution to third-order homogeneous linear differential equations with constant coefficients
y^(3)+py"+qy+ry=0
1. Write the characteristic equation into ^(3)+p into ^(2)+q into +r=0
Write the general solution to the equation
①When input 1≠enter 2≠enter 3 (single real root), Y=C1e^(enter 1x)+C2e^(enter 2x)+C3e^(enter 3x)
② When entering 1 = entering 2 ≠ entering 3 (double real root), Y = [C1 + C2x] e^ (entering 1x) + C3e^ (entering 3x)
③ When entering 1 = entering 2 = entering 3 (triple real root), Y = [C1+C2+C3x^2]e^ (entering 1x)
④ Enter 1, 2=α±阝i, enter 3∈ single real root, Y=e^(αx)〈C1cos阝x+C2sin阝x〉+C3e\(enter 3x)
6. Euler’s equation
x^ny^(n)+a1x^(n-1)y^(n-1)+…+an-1xy+any=f(x)
1. Let x=e^t d/dt=D, d^2/dt^2=D^2,…, d^n/dt^n=D^n 2. Then xy'=Dy=dy/dt x^ny^(n)=D(D-1)…(D-n+1)y 3. Substituting into the original equation, it can be transformed into a high-order constant coefficient linear differential equation.