Ordinary Differential Equations/First Order Linear 4

1)

$y^{\prime }+3y=sin(x)$

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

$\int 3dx=3x+C$

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

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

Step 2: Multiply through

$e^{3x}{y}^{\prime }+e^{3x}3y=e^{3x}sin(x)$

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)

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

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

$(x+3)y^{\prime }+(x+3)y=(x+3)(7x^2+4x)$

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

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

3)

$(x^4{e}^x-2mx{y}^2)dx+2m{x}^{2}ydy$

Step 1: Rearrange

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

Step 2: Substitute $z=y^2⟹\frac{dz}{dx}=2y\frac{dy}{dx}$

$\frac{dz}{dx}-2\frac{z}{x}=-x^2{e}^x$

Step 3: Find $e^{\int P(x)dx}$

Integrating Factor$=\frac{1}{x^2}$

Step 4: Solve for y

$y(x)=x^2\int \frac{-x^2{e}^{x}dx}{m{x}^2}=x^2\int \frac{-e^x}{m}dx=\frac{-x^2{e}^x}{m}+C{x}^2$ Template:BookCat