Ordinary Differential Equations/First Order Linear 4

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1)

y'+3y=sin(x)\,\!

Step 1: Find e^{\int P(x)dx}

\int 3dx=3x+C

e^{\int P(x)dx}=Ce^{3x}

Letting C=1, we get e^{3x}


Step 2: Multiply through

e^{3x}y'+e^{3x}3y=e^{3x}sin(x)\,\!

Step 3: Recognize that the left hand is \frac{d}{dx} e^{\int P(x)dx}y

\frac{d}{dx} e^{3x}y=e^{3x}sin(x)

Step 4: Integrate

\int (\frac{d}{dx} e^{3x}y)dx=\int e^{3x}sin(x)dx

e^{3x}y=\frac{e^{3x}(3sin(x)-cos(x))}{10}+C

Step 5: Solve for y

y=\frac{3sin(x)-cos(x)}{10}+\frac{C}{e^{3x}}


2)

y'+\frac{1}{x+3}y=7x^2+4x

Step 1: Find e^{\int P(x)dx}

\int \frac{dx}{x+3}=ln(x+3)+C

e^{\int P(x)dx}=Cx+3C

Letting C=1, we get x+3


Step 2: Multiply through

(x+3)y'+(x+3)y=(x+3)(7x^2+4x)\,\!

Step 3: Recognize that the left hand is \frac{d}{dx} e^{\int P(x)dx}y

\frac{d}{dx} (x+3)y=(x+3)(7x^2+4x)

Step 4: Integrate

\int (\frac{d}{dx} (x+3)y)dx=\int (x+3)(7x^2+4x)dx

(x+3)y=\frac{7x^4}{4}+\frac{25x^3}{3}+6x^2+C

Step 5: Solve for y

y=\frac{\frac{7x^4}{4}+\frac{25x^3}{3}+6x^2+C}{x+3}