We have not discussed methods by which systems of first-order differential equations can be solved. Nevertheless, systems such as (2) can be solved with no knowledge other than how to solve a single linear first-order equation. Find a solution of (2) subject to the initial conditions x(0)= x_0, y(0)=0, z(0)=0. dx/dt=-λ_1 x dy/dt=λ_1 x-λ_2 y dz/dt=λ_2 y #DifferentialEquations #Calculus #Techniques #SystemOfEquations

Potassium-40 Decay The chemical element potassium is a soft metal that can be found extensively throughout the Earth's crust and oceans. Although potassium occurs naturally in the form of three isotopes, only the isotope potassium-40 (K-40) is radioactive. This isotope is also unusual in that it decays by two different nuclear reactions. Over time, by emitting beta particles a great percentage of an initial amount K₀ of K-40 decays into the stable isotope calcium-40 (Ca-40), whereas by electron capture a smaller percentage of K₀ decays into the stable isotope argon-40 (Ar-40). Because the rates at which the amounts C(t) of Ca-40 and A(t) of Ar-40 increase are proportional to the amount K(t) of potassium present, and the rate at which K(t) decays is also proportional to K(t), we obtain the system of linear first-order equations dC/dt=λ_1 K dA/dt=λ_2 K dK/dt=-(λ_1+λ_2)K where λ₁ and λ₂ are positive constants of proportionality. By proceeding as in Problem 1, we can solve the mathematical model described. (a) From the last equation in the given system of differential equations, find K(t) if K(0) = K₀. Then use K(t) to find C(t) and A(t) from the first and second equations. Assume that C(0) = 0 and A(0) = 0. (b) It is known that λ₁ = 4.7526 × 10⁻¹⁰ and λ₂ = 0.5874 × 10⁻¹⁰. Find the half-life of K-40. (c) Use C(t) and A(t) found in part (a) to determine the percentage of an initial amount K₀ of K-40 that decays into Ca-40 and the percentage that decays into Ar-40 over a very long period of time. #CalculusHelp #DiffernetialEquations #SystemOfEquations

The sine integral function is defined as Si(x)=∫_0^x sint/t dt Where the integrand is defined to be 1 at x = 0. Express the solution of the initial-value problem x^3 dy/dx+2x^2 y=10sinx ,y(1)=0 In terms of Si(x) #DifferentialEquations #SineIntegral #Calculus #Techniques

Calculus Help: Find an explicit solution of the given initial-value problem dy/dx=ye^(-x^2 ),y(4)=1 #CalculusHelp #DummyVariable #DifferentialEquations

Calculus: Linear Ordinary Differential Equation: L di/dt+Ri=E,i(0)=i_0 ,L,R,E,i_0 constants #DifferentialEquations #LinearODE #CalculusHelp

Find the solution of differential equation dy/dx+2y=f(x),y(0)=0,where f(x)={ (1,0≤x≤3@0, x≫3) #DifferentialEquations #PiecewiseFunctions #Calculus

Calculus Help: Differential Equations - Linear ODE - Integrating Factor - y'=sinx (y secx-2) #DifferentialEquations #LinearODE #ODE #CalculusHelp

Calculus Help: Differential Equations - Exact - y'=(2y^2-3x^3)/(1-4xy) - Techniques #DifferentialEquations #Exact #CalculusHelp #Techniques

Calculus Help: Differential Equations - Homogeneous - x y^4 dy+(5x^5-2y^5 ) dx=0 #DifferentialEquations #CalculusHelp #Homogeneous

Calculus Help: Homogeneous Differential Equations: (x-4y)dx+(x+2y)dy=0,y=1,x=1 #Homogeneous #DifferenitalEquations #Calculus

Differential Equation Integrating Factor by Inspection: (x^2 y + y^3- y)dx + (x^3+xy^2+x)dy=0 #DifferentialEquations #Inspection #Calculus

Calculus Differential Equations - Integrating Factor by Inspection: (x^3+xy^2-y)dx+(y^3+x^2 y+x)dy=0 Calculus Differential Equations - Integrating Factor by Inspection: (x^3+xy^2-y)dx+(y^3+x^2 y+x)dy=0 #Calculus #DifferentialEquations #Inspection #Technqiues

Here is the technique to solve this differential equation and how to find them in step-by-step Calculus Help: Separable Differential Equations - y^2 (y^2+1)dx+x (y^2-1) dy=0 #SeparableDifferentialEquations #DifferentialEuqations

Separable Differential equations - y=2xy'-2y'+1 - Techniques #separabledifferentialequations #differentialequation

(e^t y+t e^t y) dt+(t e^t+2)dy=0 ,y(0)=-1 #SeparableDifferentialEquations #DifferentialEquations #CalculusHelp #Techniques

Calculus Help: Separable Differential Equations: √(4+x^2 ) y^'-xy^2=0 ,y(0)=-1/4 #SeparableDifferentialEquations #DifferentialEquations

An object of mass m is dropped into the sea at the surface. It is subject to a constant gravitational acceleration g and water resistance. (a) (4 points) If the water resistance is proportional to the object's velocity v(t) with coefficient b ≫ 0, so that Newton's second law gives b dv/dt =g- b/m v show that the object's velocity as a function of time t is given by v(t) = mg/b (1-e^(-bt/m)). We take downward to be the positive direction and the object is at rest when it is dropped at t=0. (b) (4 points) If instead the water resistance is quadratic in the velocity, so that Newton's second law gives dv/dt =g- b/m v^2 dt m dv dt find the object's velocity as a function of time t, again assuming that the object is dropped from rest. [Hint: Write Newtons' law as = g(1 − k^2v^2) where k = √b/mg. Then the algebra is a little less messy. After obtaining the final answer in terms of k, remember to replace k by √b/mg.] (c) (2 points) If the object could fall forever under these conditions without hitting the bottom, show that as t→ ∞ the velocity in part (a) approaches terminal velocity v → mg/b. What terminal velocity does the velocity in part (b) approach? #CalculusHelp #Integrals #differentialequations #PartialFractions

Calculus Help: Separable Differential Equations - (1+y^2 )dx+(1+x^2 )dy=0 #DifferentialEquations #Calculus #Separable #Techniques #CalculusHelp

Bernoulli Differential Equations: y'+xy=xy^2 , x^2 y'+2xy=y^4 , y'+y=y^4 , y'+1/x y=1/y - SOLVED!!! #Bernoulli #BernoulliDifferentialEquations #DifferentialEquations #Calculus #Techniques

Calculus Help: Bernoulli's Equations - x^3 y'+x^2 y=2y^(-4/3) - Techniques - SOLVED!!! #Bernoulli #BernoulliEquation #DifferentialEquations

x^2 y'+2xy=5y^3 Bernoulli' s Equation #BernoulliEquation #Bernoulli #DifferentialEquations #LinearODE

Laplace Transformation - System of Differential Equations: dx/dt+x-2y=0, dy/dt+x+4y=0, x(0)=1,y(0)=1 #LaplaceTransformation #DifferentialEquations #Calculus #Techniques #ToanCaoCap #ToanGiaiTich

Laplace Transformation-f(t)=3t^3-5t^2+6, f(t)=6e^5t-5t, f(t)=cos⁡(3t)+3e^2t, f(t)=3cos⁡(2t)-sin⁡(4t) #LaplaceTransformation #ToanCaoCap #ToanGiaiTich

Toán Cao Cấp Đại Học: Tìm biến đổi Laplace L{(e^(-t)+1)^2 cos⁡(2t)} - Laplace Transformation #LaplaceTransformation #ToanCaoCap #GiaiTich

y''-4y^'+4y=7sinx-cosx - Complementary Solutions and Particular Solutions - Differential Equations #DifferentialEquations #Calculus #Techniques #Solutions #ComplementarySolutions #ParticularSolutions

Calculus Help: y''-4y'+4y=0 - Complementary Solutions - Differential Equations - Techniques #ComplementarySolutions #DifferentialEquations #CalculusHelp

Here is the technique to solve this question and how to find them in step-by-step #Integral #DifferentialEquations #Calculus

Here is the technique to solve this differential equations and how to find them in step-by-step #DifferentialEquations #Calculus #Formula #Homogeneous

Complementary and Particular: y''-4y'+4y=2+8x-4x^2- Undetermined Coefficients -Differential Equation #DifferentialEquations #ComplementarySolutions #UndeterminedCoefficients

Calculus: Complementary and Particular Solutions: y'' - y' - 2y = e^(3x) - Differential Equations #DifferentialEquations #ComplementaryAndParticularSolutions #UndeterminedCoefficients

Undetermined Coefficients: Differential Equations -y''+9y=2cos2x- Complementary/Particular Solutions #DifferentialEquations #UndeterminedCoefficients #CalculusHelp

Here is the technique to solve this differential equations and how to find them in step-by-step #DifferentialEquations #IntegratingFactor #LinearODE #OrdinaryDifferentialEquations

Here is the technique to solve differential equations by Reduction Variables - y'=(x+y)/(1-3x-3y) - Techniques - SOLVED!!! #DifferentialEquations #Calculus #Formula #Techniques

Separable Differential Equations - y'=y/(x-√x) ,y(4)=e - Techniques Here is the technique to solve this differential equation and how to find them in step-by-step #Integral #Integrations #DifferentialEquations #Calculus

Here is the technique to solve this differential equation and how to find them in step-by-step #DifferentialEquations #ExactDifferentialEquations

Here is the technique to solve this question related to exact differential equation and how to find them in step-by-step #DifferentialEquations #Exact #ExactDifferentialEquations

Here is the technique to solve this differential equation and how to find them in step-by-step #DifferentialEquations #Calculus #ExactDifferentialEquations

Here is the technique to solve this differential equation and how to find them in step-by-step #DifferentialEquations #CalculusHelp #IntegratingFactor #Bernoulli

Here is the technique to solve the question about differential equations and how to find them in step-by-step #Techniques #IntegratingFactor #LinearODE #Calculus

Here is the technique to solve this question and how to find them in step-by-step #DifferentialEquations #Calculus #Exact #ExactDifferentialEquations

Here is the technique to solve this differential equation and how to find them in step-by-step #DifferentialEquations #Calculus #CalculusHelp #Techniques

Here is the technique to solve this integration factor and how to solve the linear ODE #LinearODE #IntegratingFactor #DifferentialEquations

Here is the basic way to do elimination of the arbitrary constants for differential equations #DifferentialEquations #ArbitraryConstants #Techniques #Calculus

Here is the technique to find all solutions about differential equations and how to find it in step-by-step #DifferentialEquations #Techniques #Solutions #Answered #Homogeneous #Techniques

Here is the technique to solve this question for separable differential equations and how to find them in step-by-step Calculus Help: Separable Differential Equations - dy/dx=(y^2-1)/(x^2-1) ,y(2)=2 - Techniques

Here is the technique to solve the question related to Laplace Transformation and how to find it

Here is the technique to solve this question and how to find them in step-by-step #Techniques #DifferentialEquations #Answers #Formula #calculus

Here is the technique to solve this question Homogeneous Differential Equation - dy/dx=(2xy+3y^2)/(x^2+2xy) #Homogeneous #DifferentialEquations #Solutions

Here is the technique to solve this question and how to find them in step-by-step with Differential Equations #Calculus #Integrals #Integration

Here is the technique to solve this question and how to find them in step-by-step for xy dx-(x^2+3y^2 ) dy=0 - Techniques #Techniques #Solutions #DifferentialEquations

Here is the technique to solve this question and how to find them in step-by-step: Calculus Help: Differential Equations - dy/dx=(3x+2y)/(3x+2y+2) ,y(-1)=-1 - Reduction Variable #DifferentialEquations #IntegratingFactor #Techniques #Solutions

Here is the technique to solve this differential equations and how to find them in step-by-step #DifferentialEquations #Calculus #Solutions

Here is the technique to solve this question and how to find them in step-by-step Differential Equation - Transform into Homogeneous - dy/dx=(x+y+4)/(x-y-6)

Here is the technique to solve this differential equation and how to find them in step-by-step (x+1) dy/dx+y=lnx,y(1)=10 - Techniques - Integrating Factor #IntegratingFactor #Techniques #Solutions

Here is the technique to solve this differential equation and how to solve them #DifferentialEquation #Calculus #Formula

Here is the technique to solve this question and how to find them in step-by-step #Techniques #Solutions #Answers #DifferentialEquations #Equations

Here is the technique to solve this question related to differential equation and how to find them in here #Differential #Equation #Techniques

Here is the technique to solve this question and how to find them in step-by-step #Techniques #Limits #Formula #Solutions #Answered

Here is the technique to solve this question and how to find them in step-by-step #Calculus #DifferentialEquations

Here is the technique to solve this equation related to differential equations and how to find them in step-by-step

Here is the technique to find the exact differential equation and how to find them in step-by-step

Here is the technique to solve this question and how to find it. Calculus Help: Differential Equation: x dy/dx+2y=x^3

Here is the technique to solve this question: Calculus Help: Differential Equations - Homogeneous - (x^2+y^2 )dx-xydy=0 - Techniques

Here is the technique to solve this differential equation and how to solve #Calculus #Formula #Solutions

Here is the technique to solve the separable differential equations and how to find the in step-by-step #SeparableDifferentialEquations #Calculus #BasicIntegrals

Here is the technique to solve this question and how to find them in step-by-step #Techniques #DifferentialEquations #Equations #Answers #Homogeneous