Ordinary Differential Equations/First Order Linear 4

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

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

3)

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

Step 1: Rearrange

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

Step 2: Substitute $z=y^{2}\implies {\dfrac {dz}{dx}}=2y{\dfrac {dy}{dx}}$

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

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

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

Step 4: Solve for y

$y(x)=x^{2}\int {\frac {-x^{2}e^{x}dx}{mx^{2}}}=x^{2}\int {\frac {-e^{x}}{m}}dx={\frac {-x^{2}e^{x}}{m}}+Cx^{2}$