Integrating Factor
Differential Equation

87266 Let the solution curve of the differential equation
\(x \frac{d y}{d x}-y=\sqrt{y^{2}+16 x^{2}}, y(1)=3 \text { be } y=y(x)\)
Then \(y(2)\) is equal to :

1 15
2 11
3 13
4 17
Differential Equation

87267 What is the solution of the differential equation \(\left(x+2 y^{3}\right) \frac{d y}{d x}=y ?\)

1 \(x=y^{3}+c y\)
2 \(x^{2}=y^{3}+c y^{2}\)
3 \(x^{2}=y^{3}+c y\)
4 None of the above
Differential Equation

87268 Let \(y=y(x)\) be a solution curve of the differential \(\left(1-x^{2} y^{2}\right) d x=y d x+x d y\). If the line \(x=1\) intersects the curve \(y=y(x)\) at \(y\) \(=2\) and the line \(x=2\) intersects the curve \(y=\) \(y(x)\) at \(y=\alpha\), then a value of \(\alpha\) is

1 \(\frac{3 \mathrm{e}^{2}}{2\left(3 \mathrm{e}^{2}-1\right)}\)
2 \(\frac{3 \mathrm{e}^{2}}{2\left(3 \mathrm{e}^{2}+1\right)}\)
3 \(\frac{1-3 \mathrm{e}^{2}}{2\left(3 \mathrm{e}^{2}+1\right)}\)
4 \(\frac{1+3 \mathrm{e}^{2}}{2\left(3 \mathrm{e}^{2}-1\right)}\)
Differential Equation

87269 Let \(y=y(x)\) be the solution of the differential equation \(x\left(1-x^{2}\right) \frac{d y}{d x}+\left(3 x^{2} y-y-4 x^{3}\right)=0, x>\) 1 , with \(y(2)=-2\). Then \(y(3)\) is equal to

1 -18
2 -12
3 -6
4 -3
Differential Equation

87266 Let the solution curve of the differential equation
\(x \frac{d y}{d x}-y=\sqrt{y^{2}+16 x^{2}}, y(1)=3 \text { be } y=y(x)\)
Then \(y(2)\) is equal to :

1 15
2 11
3 13
4 17
Differential Equation

87267 What is the solution of the differential equation \(\left(x+2 y^{3}\right) \frac{d y}{d x}=y ?\)

1 \(x=y^{3}+c y\)
2 \(x^{2}=y^{3}+c y^{2}\)
3 \(x^{2}=y^{3}+c y\)
4 None of the above
Differential Equation

87268 Let \(y=y(x)\) be a solution curve of the differential \(\left(1-x^{2} y^{2}\right) d x=y d x+x d y\). If the line \(x=1\) intersects the curve \(y=y(x)\) at \(y\) \(=2\) and the line \(x=2\) intersects the curve \(y=\) \(y(x)\) at \(y=\alpha\), then a value of \(\alpha\) is

1 \(\frac{3 \mathrm{e}^{2}}{2\left(3 \mathrm{e}^{2}-1\right)}\)
2 \(\frac{3 \mathrm{e}^{2}}{2\left(3 \mathrm{e}^{2}+1\right)}\)
3 \(\frac{1-3 \mathrm{e}^{2}}{2\left(3 \mathrm{e}^{2}+1\right)}\)
4 \(\frac{1+3 \mathrm{e}^{2}}{2\left(3 \mathrm{e}^{2}-1\right)}\)
Differential Equation

87269 Let \(y=y(x)\) be the solution of the differential equation \(x\left(1-x^{2}\right) \frac{d y}{d x}+\left(3 x^{2} y-y-4 x^{3}\right)=0, x>\) 1 , with \(y(2)=-2\). Then \(y(3)\) is equal to

1 -18
2 -12
3 -6
4 -3
Differential Equation

87266 Let the solution curve of the differential equation
\(x \frac{d y}{d x}-y=\sqrt{y^{2}+16 x^{2}}, y(1)=3 \text { be } y=y(x)\)
Then \(y(2)\) is equal to :

1 15
2 11
3 13
4 17
Differential Equation

87267 What is the solution of the differential equation \(\left(x+2 y^{3}\right) \frac{d y}{d x}=y ?\)

1 \(x=y^{3}+c y\)
2 \(x^{2}=y^{3}+c y^{2}\)
3 \(x^{2}=y^{3}+c y\)
4 None of the above
Differential Equation

87268 Let \(y=y(x)\) be a solution curve of the differential \(\left(1-x^{2} y^{2}\right) d x=y d x+x d y\). If the line \(x=1\) intersects the curve \(y=y(x)\) at \(y\) \(=2\) and the line \(x=2\) intersects the curve \(y=\) \(y(x)\) at \(y=\alpha\), then a value of \(\alpha\) is

1 \(\frac{3 \mathrm{e}^{2}}{2\left(3 \mathrm{e}^{2}-1\right)}\)
2 \(\frac{3 \mathrm{e}^{2}}{2\left(3 \mathrm{e}^{2}+1\right)}\)
3 \(\frac{1-3 \mathrm{e}^{2}}{2\left(3 \mathrm{e}^{2}+1\right)}\)
4 \(\frac{1+3 \mathrm{e}^{2}}{2\left(3 \mathrm{e}^{2}-1\right)}\)
Differential Equation

87269 Let \(y=y(x)\) be the solution of the differential equation \(x\left(1-x^{2}\right) \frac{d y}{d x}+\left(3 x^{2} y-y-4 x^{3}\right)=0, x>\) 1 , with \(y(2)=-2\). Then \(y(3)\) is equal to

1 -18
2 -12
3 -6
4 -3
Differential Equation

87266 Let the solution curve of the differential equation
\(x \frac{d y}{d x}-y=\sqrt{y^{2}+16 x^{2}}, y(1)=3 \text { be } y=y(x)\)
Then \(y(2)\) is equal to :

1 15
2 11
3 13
4 17
Differential Equation

87267 What is the solution of the differential equation \(\left(x+2 y^{3}\right) \frac{d y}{d x}=y ?\)

1 \(x=y^{3}+c y\)
2 \(x^{2}=y^{3}+c y^{2}\)
3 \(x^{2}=y^{3}+c y\)
4 None of the above
Differential Equation

87268 Let \(y=y(x)\) be a solution curve of the differential \(\left(1-x^{2} y^{2}\right) d x=y d x+x d y\). If the line \(x=1\) intersects the curve \(y=y(x)\) at \(y\) \(=2\) and the line \(x=2\) intersects the curve \(y=\) \(y(x)\) at \(y=\alpha\), then a value of \(\alpha\) is

1 \(\frac{3 \mathrm{e}^{2}}{2\left(3 \mathrm{e}^{2}-1\right)}\)
2 \(\frac{3 \mathrm{e}^{2}}{2\left(3 \mathrm{e}^{2}+1\right)}\)
3 \(\frac{1-3 \mathrm{e}^{2}}{2\left(3 \mathrm{e}^{2}+1\right)}\)
4 \(\frac{1+3 \mathrm{e}^{2}}{2\left(3 \mathrm{e}^{2}-1\right)}\)
Differential Equation

87269 Let \(y=y(x)\) be the solution of the differential equation \(x\left(1-x^{2}\right) \frac{d y}{d x}+\left(3 x^{2} y-y-4 x^{3}\right)=0, x>\) 1 , with \(y(2)=-2\). Then \(y(3)\) is equal to

1 -18
2 -12
3 -6
4 -3