Ordinary Differential Equations/First Order Linear 4

1)

${\displaystyle y'+3y=sin(x)\,\!}$

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

${\displaystyle \int 3dx=3x+C}$

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

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

Step 2: Multiply through

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

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

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

Step 4: Integrate

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

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

Step 5: Solve for y

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

2)

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

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

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

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

Letting C=1, we get ${\displaystyle x+3}$

Step 2: Multiply through

${\displaystyle (x+3)y'+(x+3)y=(x+3)(7x^{2}+4x)\,\!}$

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

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

Step 4: Integrate

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

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

Step 5: Solve for y

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

3)

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

Step 1: Rearrange

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

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

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

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

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

Step 4: Solve for y

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