Linear Differential Equations

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Linear (First Order) Differential Equations
Key Identifier
Linear first order DEs can be put in the form dydx+y⋅P(x)=Q(x)\frac{dy}{dx}+y\cdot P\left(x\right)=Q\left(x\right)dxdy​+y⋅P(x)=Q(x)

Strategies for Solving Linear DEs
*Remember to add a constant after the integral
*This constant will also be multiplied by 1R\frac{1}{R}R1​

IVP
If an initial condition is given, use it to solve for the unknown constant C.

Practice Question
Find the integrating factor that can be used to solve the linear differential equation e−x dydx+y=2xe^{-x}\ \frac{dy}{dx}+y=2xe−x dxdy​+y=2x.

Practice Question
Solve the DE dydx=x2(1+3y)\frac{dy}{dx}=x^2\left(1+3y\right)dxdy​=x2(1+3y).

y=−13+Cex3y=-\frac{1}{3}+Ce^{x^3}y=−31​+Cex3

y=x33(x+3)+Cy=\frac{x^3}{3}\left(x+3\right)+Cy=3x3​(x+3)+C

y=x33(x+3ex)+Cy=\frac{x^3}{3}\left(x+3e^x\right)+Cy=3x3​(x+3ex)+C

y=x33(x+ex3)+Cy=\frac{x^3}{3}\left(x+e^{x^3}\right)+Cy=3x3​(x+ex3)+C

I don't know

Practice Question
Solve the differential equation dydx+(tan⁡x)y=sin⁡x\frac{dy}{dx}+\left(\tan x\right)y=\sin xdxdy​+(tanx)y=sinx subject to y(2π)=π3y\left(2\pi\right)=\frac{\pi}{3}y(2π)=3π​.

y=cos⁡x[−ln⁡(cos⁡x)]+π3y=\cos x\left[-\ln\left(\cos x\right)\right]+\frac{\pi}{3}y=cosx[−ln(cosx)]+3π​

y=cos⁡x[−ln⁡(cos⁡x)+π3]y=\cos x\left[-\ln\left(\cos x\right)+\frac{\pi}{3}\right]y=cosx[−ln(cosx)+3π​]

y=sin⁡x[−ln⁡(sin⁡x)+π3]y=\sin x\left[-\ln\left(\sin x\right)+\frac{\pi}{3}\right]y=sinx[−ln(sinx)+3π​]

y=sin⁡x[−ln⁡(sin⁡x)]+π3y=\sin x\left[-\ln\left(\sin x\right)\right]+\frac{\pi}{3}y=sinx[−ln(sinx)]+3π​

I don't know

Practice Question
Solve the initial value problem  (x−3)dydx=x−3−y\left(x-3\right)\frac{dy}{dx}=x-3-y(x−3)dxdy​=x−3−y and y(4)=0y\left(4\right)=0y(4)=0.

y=1x−3[x22−3x+4]y=\frac{1}{x-3}\left[\frac{x^2}{2}-3x+4\right]y=x−31​[2x2​−3x+4]

y=1x−3[x+e4x]y=\frac{1}{x-3}\left[x+e^{4x}\right]y=x−31​[x+e4x]

y=1x−3[x−e4x]y=\frac{1}{x-3}\left[x-e^{4x}\right]y=x−31​[x−e4x]

y=1x(x−3)[x+e4x]y=\frac{1}{x\left(x-3\right)}\left[x+e^{4x}\right]y=x(x−3)1​[x+e4x]

y=1x(x−3)[x−e4x]y=\frac{1}{x\left(x-3\right)}\left[x-e^{4x}\right]y=x(x−3)1​[x−e4x]

I don't know

Wize University Calculus 2 Textbook > Differential Equations

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Linear (First Order) Differential Equations

Key Identifier

Strategies for Solving Linear DEs

IVP

Practice Question

Practice Question

Practice Question

Practice Question

Extra Practice

Practice: Linear DE IVP

Practice: Linear DE

Practice: Linear DE IVP

Practice: Linear DE IVP

Practice: Linear DE

Practice: Linear DE IVP