Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations5 Marks
Question
Solve the following differential equation : $ x d y+\left(y-x^3\right) d x=0 $
✓
Answer
The given differential equation is or $ \ \ \ \ \ x d y+\left(y-x^3\right) d x=0 $ or $\quad x \frac{d y}{d x}+y=x^3$ or $\quad \frac{d y}{d x}+\frac{1}{x} y=x^2$ $ x \frac{d y}{d x}+y-x^3=0 $ Comparing equation (1) with the linear differential equation $\frac{d y}{d x}+ P y= Q$ Here $P =\frac{1}{x}$ and $Q =x^2$ Integrating factor $ \text { I.F. }=e^{\int P d x}=e^{\int \frac{1}{x} d x}=e^{\log x}=x $ Hence the required solution of the differential equation will be : $ \begin{aligned} y \times \text { I.F. } & =\int(I . F) Q d x+C \\ \Rightarrow \quad y \times x & =\int(x) \cdot x^2 d x+C \\ \Rightarrow \quad y x & =\int x^3 d x+C \\ \Rightarrow \quad & =\frac{x^4}{4}+C \\ \Rightarrow \quad y & =\frac{1}{4} x^3+\frac{C}{x} \end{aligned} $ This is the general solution of the given differential equation.
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