Question
Solve the following differential equations:
$\frac{\text{dy}}{\text{dx}}=2\text{xy, y}(0)=1$
$\frac{\text{dy}}{\text{dx}}=2\text{xy, y}(0)=1$
✓
Answer
$\frac{\text{dy}}{\text{dx}}=2\text{xy, y}(0)=1$
$\int\frac{\text{dy}}{\text{y}}=\int2\text{x dx}$
$\log|\text{y}|=2\frac{\text{x}^2}{2}+\text{C}$
$\log|\text{y}|=\text{x}^2+\text{C}...(1)$
Put $\text{x}=0,\text{y}=1$
$\log(1)=0+\text{C}$
$0=0+\text{C}$
$\text{C}=0$
Put $\text{C}=0$ in equation (1),
$\log\text{y = x}^2$
$\text{y = e}^{\text{x}^{2}}$
$\int\frac{\text{dy}}{\text{y}}=\int2\text{x dx}$
$\log|\text{y}|=2\frac{\text{x}^2}{2}+\text{C}$
$\log|\text{y}|=\text{x}^2+\text{C}...(1)$
Put $\text{x}=0,\text{y}=1$
$\log(1)=0+\text{C}$
$0=0+\text{C}$
$\text{C}=0$
Put $\text{C}=0$ in equation (1),
$\log\text{y = x}^2$
$\text{y = e}^{\text{x}^{2}}$
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