# Worksheet: Initial Value Problems of Differential Equations

In this worksheet, we will practice solving initial value problems of differential equations.

**Q1: **

Find a particular solution for the following differential equation for which :

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**Q2: **

Find the solution for the following differential equation for :

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**Q3: **

Find the solution for the following differential equation for :

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**Q4: **

Find the solution for the following differential equation for :

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**Q5: **

Find the particular solution for the following separable differential equation: .

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**Q6: **

Find the solution for the following differential equation for :

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**Q7: **

Find the solution for the following differential equation for :

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**Q8: **

Find the solution for the following differential equation for :

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**Q9: **

For a circuit containing a capacitor and a resistor, the first-order differential equation that describes a discharging capacitor is If represents the charge within the capacitor at , find the general solution. is the capacitance of the capacitor. is the resistance of the resistor.

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**Q10: **

Suppose that and when . Find in terms of .

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**Q11: **

Find the solution of the differential equation that satisfies the initial condition .

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**Q12: **

Find the solution of the differential equation that satisfies the initial condition .

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**Q13: **

Solve the differential equation for given that .

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**Q14: **

The gradient of the tangent to a curve is . Find the equation of the curve given that the curve passes through the point .

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**Q15: **

Find the solution of the differential equation , where , that satisfies the initial condition .

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**Q16: **

Solve the differential equation for given that .

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**Q17: **

Find the solution of the differential equation that satisfies the initial condition .

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**Q18: **

Solve the differential equation for given that .

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**Q19: **

Find the solution for the following differential equation for :

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**Q20: **

Find the solution of the differential equation that satisfies the initial condition .

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**Q21: **

Find the solution for the following differential equation for :

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**Q22: **

The gradient of the tangent of a curve is and the curve passes through the point . Find the equation of the normal to the curve at the point where the -coordinate is .

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