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ODE

First Order ODE

Linear

1.\:\:y^{\prime}+y=e^{t}+11

2.\:\:y^{\prime}\cos(x)+y\sin(x)=1

3.\:\:y^{\prime}\cos(x)=y\sin(x)+\sin(2x)

4.\:\:y^{\prime}-2xy=e^{x^{2}}

5.\:\:y^{'}+y=(x+1)^2,\:y(0)=0

6.\:\:2y+(x+1)\frac{dy}{dx}=3x+3,\:y(0)=3

7.\:\:y^{'}-3y=3e^{4t},\:y(0)=5

8.\:\:y^{'}=\frac{1}{3}y-e^{-4x},\:y(0)=7

9.\:\:y^{'}=x+y,\:y(0)=1

10.\:\:x\frac{dy}{dx}+y=x,\:y(1)=1

ODE

First Order ODE

Separable

1.\:\:y^{\prime}=x^{2}y

2.\:\:\frac{dy}{dx}=y^{2}e^{-x}

3.\:\:(x^{2}+9)\frac{dy}{dx}+xy=0

4.\:\:y^{\prime}=y

5.\:\:y^{\prime}+x=xy^{2}

6.\:\:y^{\prime}=y^{2}\cos(x)+4y^{\prime}\sin^{2}(x)

7.\:\:y^{'}(x)=y(x)+2,\:y(0)=0

8.\:\:xyy^{\prime}=1+x^{2}+y^{2}+x^{2}y^{2}

9.\:\:\frac{dy}{dx}=\frac{y^2}{x^3},\:y(1)=1

10.\:\:y^{'}=-4xy^2,\:y(0)=1

ODE

First Order ODE

Bernoulli

1.\:\:y^{\prime}+y=y^{2}(\cos(x)-\sin(x))

2.\:\:x^{3}y^{\prime}=x^{2}y-y^{3}

3.\:\:xy^{\prime}+y^{2}\ln(x)+y=0

4.\:\:x^{2}\frac{dy}{dx}-2xy=3y^{4},\:y(1)=\frac{1}{2}

5.\:\:x^{2}\frac{d}{dx}y+y^{2}=xy

6.\:\:y^{\prime}=y(xy^{3}-1)

7.\:\:\frac{d}{dx}y=y-\frac{1}{2}xy^{2}

8.\:\:xy^{2}y^{\prime}+3x^{3}=y^{3}

9.\:\:xy^{2}\frac{d}{dx}y=x^{2}+y^{3}

10.\:\:xy^{\prime}+y+xy^{2}=0

ODE

Second Order ODE

Real Roots

1.\:\:y^{\prime\prime}-y^{\prime}-y=0

2.\:\:\frac{d^{2}x}{dt^{2}}-64x=0

3.\:\:2y^{\prime\prime}+y^{\prime}=y

4.\:\:y^{\prime\prime}-y^{\prime}-2y=0

5.\:\:3y^{\prime\prime}-10y^{\prime}+2y=0

6.\:\:y^{\prime\prime}-4y=0

7.\:\:2y^{\prime\prime}+3y^{\prime}-2y=0

8.\:\:y^{\prime\prime}+7y^{\prime}-30y=0

9.\:\:\frac{d^{2}}{dx^{2}}y+3\frac{d}{dx}y+2y=0

10.\:\:y^{\prime\prime}+y^{\prime}=y

ODE

Second Order ODE

Complex Roots

1.\:\:y^{\prime\prime}+2y^{\prime}+2y=0

2.\:\:y^{\prime\prime}+y=0

3.\:\:y^{\prime\prime}-4y^{\prime}+13y=0

4.\:\:4y^{\prime\prime}+4y^{\prime}+5y=0

5.\:\:y^{\prime\prime}+y^{\prime}+y=0

6.\:\:y^{\prime\prime}+6y^{\prime}+13y=0,\:y(0)=0

7.\:\:y^{\prime\prime}=-2y

8.\:\:9y^{\prime\prime}=-25y

9.\:\:y^{\prime\prime}=-2y,\:y(π)=\frac{1}{2}

10.\:\:y^{\prime\prime}+6y^{\prime}+25y=0

ODE

Second Order ODE

One Root

1.\:\:0.25y^{\prime\prime}-2y^{\prime}+4y=0

2.\:\:16y^{\prime\prime}-8y^{\prime}+y=0,\:y(0)=40

3.\:\:y^{\prime\prime}=0

4.\:\:9\frac{d^{2}y}{dx^{2}}-6\frac{dy}{dx}+y=0

5.\:\:y^{\prime\prime}+2y^{\prime}+y=0

6.\:\:y^{\prime\prime}+14y^{\prime}+49y=0

7.\:\:100y^{\prime\prime}+60y^{\prime}+9y=0

8.\:\:100y^{\prime\prime}+60y^{\prime}+9y=0,\:y(1)=8

9.\:\:y^{\prime\prime}-6y^{\prime}+9y=0

10.\:\:4y^{\prime\prime}+12y^{\prime}+9y=0